Analyzing and Interpreting Graphs

1. Intercepts

Introduction

When studying graphs, one of the first things we often want to know is where the graph meets the coordinate axes.

These points are called intercepts.

Intercepts help us understand the behaviour of functions, solve equations, compare graphs, and interpret mathematical relationships in real-world situations.

For example, consider the linear function:

y = 2x − 6

The graph crosses the x-axis at (3, 0) and the y-axis at (0, −6).

These two points tell us important information:

• The x-intercept shows where the output equals zero.

• The y-intercept shows the output when the input equals zero.

In a business model, an x-intercept might represent a break-even point, while a y-intercept might represent an initial cost or starting balance.

In physics, an x-intercept on a height-time graph might represent when an object reaches ground level, while the y-intercept represents its initial height.

Intercepts are useful for linear, quadratic, exponential, polynomial, and many other types of functions.

What Is an Intercept?

An intercept is a point where a graph intersects one of the coordinate axes.

There are two main types:

• x-intercepts: Points where the graph meets the x-axis.

• y-intercepts: Points where the graph meets the y-axis.

The x-axis is the horizontal axis.

The y-axis is the vertical axis.

Every point on the x-axis has a y-coordinate of zero.

Every point on the y-axis has an x-coordinate of zero.

This gives us two essential rules:

To find an x-intercept, set y = 0.

To find a y-intercept, set x = 0.

Visualising Intercepts

2. Increasing and Decreasing Functions

Introduction

Graphs show more than the relationship between two variables. They also reveal how a quantity changes as its input increases.

For example, imagine tracking the height of a ball thrown into the air.

• At first, the ball moves upward, so its height increases.

• At its highest point, its height stops increasing.

• As the ball falls, its height decreases.

These changes can be described mathematically using increasing and decreasing functions.

Understanding where a function increases or decreases allows us to identify trends, compare graphs, locate maximum and minimum values, and interpret real-world data.

This is useful in many subjects:

• Physics: Describing changes in height, speed, and energy.

• Biology: Studying population growth and decline.

• Economics: Analysing changes in profit, revenue, and costs.

• Chemistry: Investigating how concentrations change during reactions.

• Environmental science: Interpreting changes in temperature and other measurements.

What Is an Increasing Function?

A function is increasing over an interval when its output increases as its input increases.

In simple terms:

As x increases, y increases.

Imagine moving from left to right along a graph. If the graph rises, the function is increasing.

Example: A linear function

Consider:

y = 2x + 1

Calculate some values:

As x increases from −2 to 2, y increases from −3 to 5.

Therefore, the function is increasing over its entire domain.

Mathematical definition

A function f is strictly increasing on an interval if, whenever x₁ < x₂ in that interval:

f(x₁) < f(x₂)

This means a larger input always gives a larger output.

For example:

If x₁ = 1 and x₂ = 3, then for f(x) = 2x + 1:

f(1) = 3

f(3) = 7

Since 1 < 3 and 3 < 7, these values are consistent with an increasing function.

One pair of values alone does not prove that a function is increasing over an entire interval, but it helps illustrate the idea.

What Is a Decreasing Function?

A function is decreasing over an interval when its output decreases as its input increases.

In simple terms:

As x increases, y decreases.

When moving from left to right along a graph, a decreasing function falls.

Example: A linear function

Consider:

y = −3x + 4

As x increases, y decreases.

Therefore, the function is decreasing over its entire domain.

Mathematical definition

A function f is strictly decreasing on an interval if, whenever x₁ < x₂:

f(x₁) > f(x₂)

This means a larger input always gives a smaller output.

Constant Functions

Not every function is increasing or decreasing.

Consider:

y = 5

Regardless of the value of x, the output remains 5.

The graph is a horizontal line.

It is constant over its entire domain.

In strict terminology, a constant function is neither strictly increasing nor strictly decreasing.

Some textbooks also use non-decreasing and non-increasing:

• Non-decreasing means outputs never go down as x increases.

• Non-increasing means outputs never go up as x increases.

Under these broader definitions, a constant function is both non-decreasing and non-increasing.

For these notes, increasing and decreasing will generally mean strictly increasing and decreasing unless stated otherwise.

Understanding Intervals

Many functions are not increasing or decreasing everywhere.

Instead, their behaviour changes over different parts of their domains.

An interval is a continuous section of the x-axis.

For example:

−3 < x < 2

describes all x-values between −3 and 2.

In interval notation, this is written:

(−3, 2)

The parentheses indicate that the endpoints are not included.

Common interval notation

Infinity always uses parentheses because it is not a real endpoint.

Why interval notation matters

A function may increase over one interval and decrease over another.

We need a precise way to describe where each behaviour occurs.

Identifying Increasing and Decreasing Intervals from Graphs

To identify intervals:

1. Read the graph from left to right.

2. Identify sections where the graph rises.

3. Identify sections where the graph falls.

4. Locate any turning points.

5. Record the corresponding x-values using interval notation.

Worked Example: A quadratic function

Consider:

f(x) = x²

The graph is a U-shaped parabola with vertex at (0, 0).

On the left side of the vertex, the graph falls as x increases.

On the right side, the graph rises.

Therefore:

Decreasing interval: (−∞, 0)

Increasing interval: (0, ∞)

The vertex at x = 0 is a turning point where the direction changes.

We usually use open intervals when reporting increasing and decreasing behaviour so that turning points clearly separate the intervals. For continuous functions, including the turning-point endpoint can also be mathematically valid when describing monotonicity on a closed or half-closed interval.

Increasing and Decreasing Quadratic Functions

The direction of a quadratic function depends on whether its parabola opens upward or downward.

The vertex is particularly important because it separates the increasing and decreasing intervals.

Upward-opening parabolas

Consider:

f(x) = (x − 3)² + 2

The vertex is:

(3, 2)

The graph opens upward, so it decreases until x = 3 and then increases.

• Decreasing: (−∞, 3)

• Increasing: (3, ∞)

• Minimum value: 2

Downward-opening parabolas

Consider:

g(x) = −2(x + 1)² + 8

The vertex is:

(−1, 8)

The graph opens downward, so it increases until x = −1 and then decreases.

• Increasing: (−∞, −1)

• Decreasing: (−1, ∞)

• Maximum value: 8

Comparing the two types

Worked Example: Finding intervals from standard form

Consider:

f(x) = 2x² − 8x + 5

The x-coordinate of the vertex is:

x = −b/(2a)

Here:

a = 2, b = −8

Substitute:

x = −(−8)/(2 × 2)

x = 8/4

x = 2

Because a is positive, the parabola opens upward.

Therefore:

Decreasing: (−∞, 2)

Increasing: (2, ∞)

To calculate the minimum:

f(2) = 2(2)² − 8(2) + 5

f(2) = 8 − 16 + 5

f(2) = −3

The minimum occurs at:

(2, −3)

Increasing and Decreasing Linear Functions

For a linear function:

y = mx + c

the gradient m determines its behaviour.

• If m > 0, the function is increasing.

• If m < 0, the function is decreasing.

• If m = 0, the function is constant.

Worked Example

Compare:

A: y = 4x + 2

B: y = −2x + 7

C: y = 5

Function A has gradient 4, so it is increasing everywhere.

Function B has gradient −2, so it is decreasing everywhere.

Function C has gradient 0, so it is constant everywhere.

Both A and B have domains consisting of all real numbers.

Therefore:

• A is increasing on (−∞, ∞).

• B is decreasing on (−∞, ∞).

• C is constant on (−∞, ∞).

Comparing rates of change

Consider:

f(x) = 2x + 1

g(x) = 5x + 1

Both functions are increasing.

However, g(x) increases more rapidly because its gradient is larger.

For each increase of 1 in x:

• f(x) increases by 2.

• g(x) increases by 5.

A function can therefore be increasing without increasing at the same rate as another function.

Increasing and Decreasing Absolute Value Functions

Consider the parent function:

f(x) = |x|

Its graph has a V shape.

The vertex is (0, 0).

To the left of the vertex, the function decreases.

To the right, it increases.

Therefore:

• Decreasing: (−∞, 0)

• Increasing: (0, ∞)

Worked Example: Transformed absolute value function

Consider:

g(x) = −3|x − 2| + 5

The vertex is:

(2, 5)

The negative coefficient reflects the graph across the x-axis.

It becomes an upside-down V.

Therefore:

Increasing: (−∞, 2)

Decreasing: (2, ∞)

The function has a maximum value of 5.

Increasing and Decreasing Exponential Functions

Exponential functions are often used to model growth and decay.

Consider:

f(x) = 2ˣ

As x increases, the output increases.

For example:

The function is increasing on:

(−∞, ∞)

Now consider:

g(x) = (1/2)ˣ

The function is decreasing on:

(−∞, ∞)

General rule

For:

y = bˣ

where b > 0 and b ≠ 1:

• If b > 1, the function is increasing.

• If 0 < b < 1, the function is decreasing.

A vertical reflection can reverse these trends.

For example:

y = −2ˣ

is decreasing because it is the reflection of an increasing exponential function across the x-axis.

Functions with Several Increasing and Decreasing Intervals

Some functions change direction more than once.

Consider:

f(x) = x³ − 3x

This cubic function has two turning points.

The turning points are:

• A local maximum at (−1, 2).

• A local minimum at (1, −2).

3. Maximums and Minimums

Introduction

Many real-world situations involve finding the greatest or smallest possible value of a quantity.

For example:

• What is the maximum height reached by a ball thrown into the air?

• What is the minimum cost of producing a product?

• At what point does a business earn its greatest profit?

• What is the lowest temperature recorded during a day?

• How can an engineer design a structure using the least amount of material?

These questions involve maximums and minimums, also known as extrema.

In mathematics, extrema are important features of functions because they help us understand the behaviour of graphs, identify turning points, compare different models, and solve optimisation problems.

A maximum is a greatest value, while a minimum is a smallest value. However, we must distinguish between values that are greatest or smallest over an entire domain and those that are greatest or smallest only compared with nearby values.

Understanding this distinction is essential when analysing functions.

What Is a Maximum Value?

A maximum value is the greatest output a function reaches over a specified domain or interval.

Consider the quadratic function:

f(x) = −x² + 9

Its graph is a downward-opening parabola.

The highest point is:

(0, 9)

Therefore:

• The maximum value is 9.

• The maximum occurs at x = 0.

• The maximum point is (0, 9).

Notice the distinction between the maximum value and the point where it occurs.

The maximum value is a y-value.

The maximum point includes both the x-coordinate and the y-coordinate.

Maximum values and increasing or decreasing functions

For this graph:

• The function increases as x approaches 0 from the left.

• The function reaches its maximum at x = 0.

• The function decreases as x moves to the right of 0.

A maximum often occurs where a graph changes from increasing to decreasing.

What Is a Minimum Value?

A minimum value is the smallest output a function reaches over a specified domain or interval.

Consider:

f(x) = x² − 4

The graph is an upward-opening parabola.

Its lowest point is:

(0, −4)

Therefore:

• The minimum value is −4.

• The minimum occurs at x = 0.

• The minimum point is (0, −4).

  

  For this graph:

  • The function decreases until x = 0.

  • The function reaches its minimum.

  • The function then increases.

  A minimum often occurs where a graph changes from decreasing to increasing.

  Maximum and Minimum Points Versus Values

  It is important to distinguish between three related ideas.

  Consider:

  f(x) = −2(x − 3)² + 10

  The graph reaches its highest point at:

  (3, 10)

  We can describe this in three ways:

  • Maximum value: 10.

  • x-value where the maximum occurs: 3.

  • Maximum point: (3, 10).

  These are not interchangeable.

  If a question asks for the maximum value, the answer is 10.

  If it asks where the maximum occurs, the answer is x = 3.

  If it asks for the coordinates of the maximum point, the answer is (3, 10).

  Local and Absolute Extrema

  Not every maximum or minimum is the greatest or smallest value over an entire function.

  We distinguish between local extrema and absolute extrema.

  Local maximum

  A local maximum occurs when a function's value is greater than or equal to nearby values.

  The graph may rise to a peak, then fall.

  However, another part of the graph may reach an even greater value.

  Local minimum

  A local minimum occurs when a function's value is less than or equal to nearby values.

  The graph may fall into a valley, then rise.

  However, another part of the graph may reach an even smaller value.

  Absolute maximum

  An absolute maximum is the greatest output of a function over its entire specified domain.

  Absolute minimum

  An absolute minimum is the smallest output of a function over its entire specified domain.

  Type	Meaning
  Local maximum	Greater than or equal to nearby outputs
  Local minimum	Less than or equal to nearby outputs
  Absolute maximum	Greatest output over the specified domain
  Absolute minimum	Smallest output over the specified domain

  Example: A cubic function

  Consider:

  f(x) = x³ − 3x

  This function has:

  • A local maximum at (−1, 2).

  • A local minimum at (1, −2).

  However, over all real numbers, the function has no absolute maximum or absolute minimum.

  As x becomes increasingly large and positive, f(x) increases without bound.

  As x becomes increasingly negative, f(x) decreases without bound.

  

  Turning Points

  A turning point is a point where a graph changes direction.

  There are two common types:

  • Increasing to decreasing: a local maximum.

  • Decreasing to increasing: a local minimum.

  Identifying turning points visually

  When reading a graph from left to right:

  1. Find where the graph stops rising and starts falling.

  2. Mark this as a possible local maximum.

  3. Find where the graph stops falling and starts rising.

  4. Mark this as a possible local minimum.

  5. Read the coordinates of each point.

  Turning points are especially important when analysing quadratic, cubic, and higher-degree polynomial functions.

  Important distinction: Stationary points

  A stationary point is a point where the derivative is zero, meaning the tangent is horizontal.

  Some stationary points are turning points, but not all.

  For example:

  f(x) = x³

  At x = 0, the graph has a horizontal tangent.

  However, it continues increasing on both sides.

  Therefore, (0, 0) is a stationary point but not a turning point.

  Also, some turning points have sharp corners, such as the vertex of y = |x|, where the derivative is undefined.

  This distinction becomes particularly important in calculus.

  Finding Maximums and Minimums of Quadratic Functions

  Quadratic functions are especially useful for studying extrema because their graphs have exactly one turning point.

  The general form is:

  f(x) = ax² + bx + c

  where a ≠ 0.

  The sign of a determines whether the vertex is a maximum or minimum.

  • If a > 0, the parabola opens upward and has a minimum.

  • If a < 0, the parabola opens downward and has a maximum.

  The x-coordinate of the vertex is:

  x = −b/(2a)

  Once this x-value is known, substitute it into the function to find the maximum or minimum value.

  Worked Example: Finding a minimum

  Consider:

  f(x) = 2x² − 12x + 11

  Step 1: Identify the coefficients

  a = 2

  b = −12

  c = 11

  Because a is positive, the parabola opens upward.

  Therefore, it has a minimum.

  Step 2: Calculate the x-coordinate of the vertex

  x = −b/(2a)

  x = −(−12)/(2 × 2)

  x = 12/4

  x = 3

  Step 3: Calculate the minimum value

  Substitute x = 3:

  f(3) = 2(3)² − 12(3) + 11

  f(3) = 18 − 36 + 11

  f(3) = −7

  Answer:

  • Minimum value: −7

  • Minimum occurs at x = 3

  • Minimum point: (3, −7)

  • Axis of symmetry: x = 3

  The function decreases on (−∞, 3) and increases on (3, ∞).

  Worked Example: Finding a maximum

  Consider:

  g(x) = −3x² + 18x − 14

  Because a = −3, the parabola opens downward.

  The vertex occurs at:

  x = −18/[2(−3)]

  x = 3

  Substitute:

  g(3) = −3(3)² + 18(3) − 14

  g(3) = −27 + 54 − 14

  g(3) = 13

  Answer:

  • Maximum value: 13

  • Maximum occurs at x = 3

  • Maximum point: (3, 13)

  The function increases on (−∞, 3) and decreases on (3, ∞).

  Finding Extrema Using Vertex Form

  Quadratic functions can also be written in vertex form:

  f(x) = a(x − h)² + k

  The vertex is:

  (h, k)

  This makes it easy to identify the extremum without calculating −b/(2a).

  Worked Example: Maximum

  Consider:

  f(x) = −4(x − 5)² + 20

  The vertex is:

  (5, 20)

  Since a = −4, the parabola opens downward.

  Therefore:

  Maximum value = 20 at x = 5

  Worked Example: Minimum

  Consider:

  g(x) = 0.5(x + 2)² − 6

  Rewrite mentally as:

  g(x) = 0.5(x − (−2))² − 6

  The vertex is:

  (−2, −6)

  Since a = 0.5 is positive, the parabola opens upward.

  Therefore:

  Minimum value = −6 at x = −2

  Why vertex form is useful

  Vertex form makes three important properties immediately visible:

  • The horizontal position of the turning point.

  • The maximum or minimum value.

  • Whether the parabola opens upward or downward.

  It is particularly useful in real-world optimisation problems.

  Finding Extrema by Completing the Square

  Sometimes a quadratic is given in standard form, but we want to rewrite it in vertex form.

  This can be done by completing the square.

  Worked Example

  Find the minimum of:

  f(x) = x² + 8x + 11

  Start with:

  f(x) = x² + 8x + 11

  Half of 8 is 4.

  Square 4 to get 16.

  Rewrite:

  f(x) = (x² + 8x + 16) − 16 + 11

  Simplify:

  f(x) = (x + 4)² − 5

  The vertex is:

  (−4, −5)

  Because the squared expression cannot be negative:

  (x + 4)² ≥ 0

  Therefore:

  (x + 4)² − 5 ≥ −5

  The smallest possible output is −5.

  Minimum value = −5 at x = −4

  Completing the square does more than locate the vertex. It also explains algebraically why the value is a minimum.

  Maximums and Minimums on Restricted Domains

  A function's maximum and minimum values depend on the domain being considered.

  For example:

  f(x) = x²

  Over all real numbers, the function has:

  • Absolute minimum: 0 at x = 0.

  • No absolute maximum.

  However, suppose the domain is restricted to:

  −2 ≤ x ≤ 3

  Now the function has both an absolute minimum and an absolute maximum.

  Calculate the important values:

  f(−2) = 4

  f(0) = 0

  f(3) = 9

  Therefore:

  • Absolute minimum: 0 at x = 0.

  • Absolute maximum: 9 at x = 3.

  The maximum occurs at an endpoint, not at a turning point.

  Worked Example: A restricted quadratic

  Find the absolute maximum and minimum of:

  f(x) = −x² + 4x + 1

  on:

  0 ≤ x ≤ 5

  Step 1: Find the vertex

  a = −1, b = 4

  x = −4/[2(−1)]

  x = 2

  Step 2: Evaluate the vertex

  f(2) = −4 + 8 + 1 = 5

  Step 3: Evaluate the endpoints

  f(0) = 1

  f(5) = −25 + 20 + 1 = −4

  Step 4: Compare

  x	f(x)
  0	1
  2	5
  5	−4

  Therefore:

  Absolute maximum = 5 at x = 2

  Absolute minimum = −4 at x = 5

  This example shows why checking only the vertex can give an incomplete answer.

  For a continuous quadratic on a closed interval, evaluating the vertex when it lies inside the interval and evaluating both endpoints is sufficient to find its absolute extrema.

  When a Maximum or Minimum Does Not Exist

  Not every function reaches a greatest or smallest value.

  Example: A linear function

  Consider:

  f(x) = 3x + 2

  over all real numbers.

  As x increases, f(x) increases without bound.

  As x decreases, f(x) decreases without bound.

  Therefore, the function has neither an absolute maximum nor an absolute minimum.

  Example: An exponential function

  Consider:

  f(x) = 2ˣ

  The function is always positive.

  As x approaches negative infinity, its outputs approach zero.

  However, the function never actually reaches zero.

  Therefore:

  • It has no absolute minimum.

  • It has no absolute maximum.

  Zero is the lower bound of its range, but it is not a minimum value.

  Example: An open interval

  Consider:

  f(x) = x

  with domain:

  0 < x < 5

  The function approaches zero and five but never reaches either endpoint.

  Therefore, it has no absolute maximum or minimum on this domain.

  This demonstrates an important distinction between a value that a function approaches and one that it actually reaches.

  Finding Maximums and Minimums from Graphs

  Sometimes we are given a graph without an equation.

  In this situation, we estimate extrema visually.

  Steps

  1. Identify the domain shown on the graph.

  2. Look for peaks and valleys.

  3. Identify the x-coordinate of each peak or valley.

  4. Read the corresponding y-value.

  5. Determine whether each point is a local or absolute extremum.

  6. Check the endpoints of the graph.

  7. Use approximate values when the coordinates are not exact.

  Example: A graph with several turning points

  Suppose a graph has the following important points:

  • A local maximum at (−3, 5).

  • A local minimum at (0, −2).

  • Another local maximum at (4, 7).

  The graph has at least two local maxima and one local minimum.

  The greatest of the listed local maximum values is 7.

  However, without knowing the entire graph and its domain, we cannot conclude that 7 is the absolute maximum.

  Other parts of the function may reach greater values.

  Estimating extrema from a graph

  When a turning point lies between grid lines, estimate its coordinates.

  For example, a maximum might appear near:

  (2.4, 8.7)

  We would write:

  Approximate maximum value ≈ 8.7 at x ≈ 2.4

  The approximation sign indicates that the coordinates were estimated rather than calculated exactly.

  Comparing Extrema Across Different Functions

  Maximum and minimum values provide useful information when comparing functions.

  Consider three quadratic functions:

  A: f(x) = −(x − 2)² + 8

  B: g(x) = −2(x − 2)² + 12

  C: h(x) = (x − 2)² − 3

  Comparing their features

  Feature	Function A	Function B	Function C
  Opens	Downward	Downward	Upward
  Vertex	(2, 8)	(2, 12)	(2, −3)
  Extremum	Maximum	Maximum	Minimum
  Extremum value	8	12	−3
  x-value of extremum	2	2	2

  Interpretation

  Functions A and B both have maximum values, but B reaches a greater maximum.

  Function C has a minimum rather than a maximum.

  All three functions have their turning points at x = 2.

  Therefore, their extrema occur at the same input value, even though their outputs and shapes differ.

  Comparing graph widths

  Function B has a larger absolute quadratic coefficient than Function A.

  This means B is narrower.

  However, the width of the parabola does not by itself determine the maximum or minimum value.

  The vertical position of the vertex determines that value.

  How Transformations Affect Extrema

  Graph transformations are closely connected to maximums and minimums.

  Start with:

  f(x) = x²

  Its minimum point is:

  (0, 0)

  Vertical translation

  g(x) = x² + 5

  The graph moves upward by 5.

  The minimum changes from 0 to 5.

  New minimum point:

  (0, 5)

  Horizontal translation

  h(x) = (x − 3)²

  The graph moves right by 3.

  The minimum value remains 0, but it occurs at x = 3.

  New minimum point:

  (3, 0)

  Reflection across the x-axis

  p(x) = −x²

  The graph is reflected vertically.

  The original minimum becomes a maximum.

  Maximum point:

  (0, 0)

  Vertical stretch

  q(x) = 4(x − 2)² + 3

  The graph is vertically stretched relative to the basic parabola, translated right by 2 and upward by 3.

  Minimum point:

  (2, 3)

  The stretch changes the graph's width but does not change the vertex when applied to the squared term alone.

  General conclusion

  For:

  y = a(x − h)² + k

  • h determines the horizontal position of the extremum.

  • k determines the extremum value.

  • The sign of a determines whether it is a maximum or minimum.

  • The magnitude of a determines the graph's vertical scaling.

  Real-World Application: Maximum Height of a Projectile

  Projectile motion is a common application of maximum values.

  Suppose a ball is thrown upward from a platform.

  Its height is modeled by:

  h(t) = −5t² + 30t + 10

  where:

  • h is height in metres.

  • t is time in seconds.

  Step 1: Find the time of maximum height

  The vertex occurs at:

  t = −b/(2a)

  a = −5, b = 30

  t = −30/[2(−5)]

  t = 3

  Step 2: Find the maximum height

  Substitute t = 3:

  h(3) = −5(3)² + 30(3) + 10

  h(3) = −45 + 90 + 10

  h(3) = 55

  Step 3: Interpret the result

  Maximum height = 55 m

  Time to maximum height = 3 s

  The ball rises for the first 3 seconds, reaches its highest point, and then begins to fall.

  This model assumes approximately constant gravitational acceleration and ignores air resistance.

  basketball scoring goal on hoop

   

   

   

  Real-World Application: Maximum Business Profit

  Businesses often try to identify the production or sales level that maximises profit.

  Suppose profit is modeled by:

  P(x) = −2x² + 120x − 1000

  where:

  • x = number of products sold.

  • P(x) = profit in dollars.

  Step 1: Determine the type of extremum

  The coefficient of x² is negative.

  Therefore, the parabola opens downward and has a maximum.

  Step 2: Find the optimal sales quantity

  x = −b/(2a)

  x = −120/[2(−2)]

  x = 30

  Step 3: Find maximum profit

  P(30) = −2(30)² + 120(30) − 1000

  P(30) = −1800 + 3600 − 1000

  P(30) = 800

  Interpretation

  Maximum modeled profit = $800

  This occurs when 30 products are sold.

  Selling fewer or more than 30 products produces a lower profit under the model.

  The model is an idealised example; actual business decisions also depend on demand, production limits, and other costs.

  Real-World Application: Minimising Costs

  Not all optimisation problems involve finding a maximum.

  Businesses and engineers often want to minimise costs.

  Suppose the operating cost of a process is:

  C(x) = 3(x − 12)² + 450

  where x represents the number of operating hours per day.

  The vertex is:

  (12, 450)

  Since the quadratic coefficient is positive, the graph opens upward.

  Therefore:

  Minimum cost = $450

  This occurs at:

  x = 12 hours

  The model suggests that operating for either fewer or more than 12 hours increases the cost.

  This might represent a process where very short operating periods create inefficiencies while very long periods lead to overtime or maintenance costs.

  Real-World Application: Maximum Area

  Optimisation is also important in geometry.

  Suppose a farmer has 40 metres of fencing to enclose a rectangular field.

  What dimensions give the greatest possible area?

  Step 1: Write the perimeter equation

  For a rectangle:

  P = 2L + 2W

  Since the perimeter is 40:

  2L + 2W = 40

  Divide by 2:

  L + W = 20

  Therefore:

  L = 20 − W

  Step 2: Write the area equation

  Area = length × width

  A = LW

  Substitute L = 20 − W:

  A(W) = W(20 − W)

  Expand:

  A(W) = −W² + 20W

  Step 3: Find the maximum

  The quadratic coefficient is negative, so the parabola opens downward.

  The vertex occurs at:

  W = −20/[2(−1)]

  W = 10

  Then:

  L = 20 − 10

  L = 10

  Step 4: Calculate the area

  A = 10 × 10

  Maximum area = 100 m²

  The greatest area occurs when the rectangle is a square measuring 10 m by 10 m.

  17 Leaning Oak Ln, Ravenel, SC 29470 | MLS #25025127 | Zillow

   

   

   

  Why this matters

  The same mathematical idea can be applied to:

  • Designing gardens.

  • Minimising packaging materials.

  • Maximising storage capacity.

  • Optimising the dimensions of containers.

  • Reducing construction costs.

  Real-World Application: Temperature Extremes

  Suppose the temperature during part of a day is modeled by:

  T(t) = −0.5(t − 14)² + 32

  where t is the hour of the day.

  The vertex is:

  (14, 32)

  The parabola opens downward.

  Therefore:

  Maximum temperature = 32°C at 14:00 (2:00 p.m.)

  If the model applies only from 8:00 to 20:00, we can also determine the minimum over this interval.

  At t = 8:

  T(8) = −0.5(8 − 14)² + 32

  T(8) = −18 + 32 = 14°C

  At t = 20:

  T(20) = −0.5(20 − 14)² + 32

  T(20) = 14°C

  Therefore, over the specified interval:

  • Maximum: 32°C at 14:00.

  • Minimum: 14°C at 8:00 and 20:00.

  This is an illustrative temperature model rather than a prediction of actual weather.

  Finding Extrema from Tables of Values

  Sometimes a function is represented by a table rather than an equation or graph.

  Consider the following measurements:

  Time (min)	Temperature (°C)
  0	18
  5	23
  10	28
  15	31
  20	29
  25	24
  30	20

  The highest recorded temperature is 31°C at 15 minutes.

  The lowest recorded temperature is 18°C at 0 minutes.

  Therefore:

  • Recorded maximum: 31°C at 15 min.

  • Recorded minimum: 18°C at 0 min.

  However, these are the extrema among the recorded measurements.

  The temperature might have reached a slightly higher or lower value between measurements.

  A continuous graph or mathematical model would be needed to investigate that possibility.

  Comparing two datasets

  Suppose two experiments produce the following results:

  Time (min)	Experiment A	Experiment B
  0	10	15
  5	18	22
  10	25	28
  15	30	26
  20	27	20

  Experiment A has a recorded maximum of 30 at 15 minutes.

  Experiment B has a recorded maximum of 28 at 10 minutes.

  Therefore:

  • Experiment A reaches a higher recorded maximum.

  • Experiment B reaches its recorded maximum earlier.

  • The experiments have different trends after reaching their peaks.

  Comparing extrema helps us understand not only how high or low a quantity becomes but also when important events occur.

  Extrema of Other Function Families

  Quadratic functions are not the only functions that can have maximums or minimums.

  Absolute value functions

  Consider:

  f(x) = |x − 3| + 2

  The graph has a V shape.

  Its vertex is:

  (3, 2)

  Therefore:

  Minimum value = 2 at x = 3

  Now consider:

  g(x) = −|x + 1| + 5

  This graph is an upside-down V.

  Its vertex is:

  (−1, 5)

  Therefore:

  Maximum value = 5 at x = −1

  These are examples of extrema at sharp turning points.

  Cubic functions

  Consider:

  f(x) = x³ − 3x

  The function has a local maximum at (−1, 2) and a local minimum at (1, −2).

  Unlike a quadratic, it has more than one turning point.

  Trigonometric functions

  Consider:

  f(x) = sin x

  The sine function repeatedly reaches:

  • Maximum value: 1.

  • Minimum value: −1.

  These extrema repeat periodically.

  For a transformed sine function:

  g(x) = 3 sin x + 2

  The amplitude is 3 and the midline is y = 2.

  Therefore:

  Maximum = 2 + 3 = 5

  Minimum = 2 − 3 = −1

  Exponential functions

  Consider:

  f(x) = 2ˣ

  Over all real numbers, this function has no maximum or minimum.

  It increases without bound and approaches zero without reaching it.

  These examples demonstrate that extrema depend on the function's shape and domain.

  An Introduction to Calculus and Extrema

  For more advanced functions, calculus provides a systematic method for locating possible extrema.

  The derivative measures the instantaneous rate of change of a function.

  It is commonly written as:

  f′(x)

  For a smooth function:

  • f′(x) > 0 means the function is increasing locally.

  • f′(x) < 0 means the function is decreasing locally.

  • f′(x) = 0 identifies a stationary point.

  A stationary point may be a local maximum, local minimum, or neither.

  Worked Example: Using the derivative

  Consider:

  f(x) = x³ − 3x

  Differentiate:

  f′(x) = 3x² − 3

  Set the derivative equal to zero:

  3x² − 3 = 0

  3x² = 3

  x² = 1

  x = −1 or x = 1

  These are the stationary points.

  Evaluate the function:

  f(−1) = (−1)³ − 3(−1)

  f(−1) = −1 + 3 = 2

  f(1) = 1 − 3 = −2

  The stationary points are:

  (−1, 2) and (1, −2)

  Now examine the sign of the derivative:

  Interval	Sign of f′(x)	Behaviour
  x < −1	Positive	Increasing
  −1 < x < 1	Negative	Decreasing
  x > 1	Positive	Increasing

  Therefore:

  • At x = −1, the function changes from increasing to decreasing: local maximum.

  • At x = 1, the function changes from decreasing to increasing: local minimum.

  Important limitation

  Finding f′(x) = 0 identifies possible extrema, but it does not guarantee an extremum.

  We must also check where the derivative changes sign or use another appropriate test.

  For absolute extrema on a closed interval, endpoints must also be considered.

  For functions with corners or other nondifferentiable points, extrema can occur even when the derivative does not exist.

  A Step-by-Step Strategy for Finding Extrema

  When solving a maximum or minimum problem, use the following approach.

  Identify the function.

  Determine the equation or graph representing the quantity.

  Identify the domain.

  Decide which input values are allowed or meaningful.

  Locate possible extrema.

  For quadratics, find the vertex.

  For graphs, identify peaks, valleys, and relevant endpoints.

  For more advanced functions, investigate stationary points and points where the derivative is undefined.

  Calculate the output values.

  Substitute the relevant x-values into the function.

  Compare the values.

  Determine which is greatest or smallest over the specified domain.

  Interpret the result.

  Explain the meaning using appropriate units.

  Check whether the result is realistic.

  Consider restrictions such as time, cost, distance, and production capacity.

  Common Mistakes

  Confusing the maximum value with the x-coordinate where it occurs

  If the maximum point is (3, 10), the maximum value is 10, not 3.

  Assuming every turning point is an absolute extremum

  A turning point may be only a local maximum or minimum.

  Ignoring the domain

  A function may have different extrema when its domain is restricted.

  Forgetting to check endpoints

  Absolute extrema on a closed interval may occur at endpoints.

  Assuming every function has a maximum and minimum

  Some functions increase or decrease without bound.

  Confusing the sign of the vertex's y-coordinate with the type of extremum

  A negative value can be a maximum, and a positive value can be a minimum.

  Assuming all stationary points are turning points

  The function y = x³ has a stationary point at the origin but no turning point there.

  Misidentifying the direction of a parabola

  For y = ax² + bx + c, a positive coefficient gives a minimum, while a negative coefficient gives a maximum.

  Ignoring units in real-world problems

  A maximum height must be expressed in units of length, while maximum profit is expressed in currency.

  Treating recorded data extrema as exact continuous extrema

  The greatest measured value may not be the greatest value reached between measurements.

  Key Terms

  • Maximum value: The greatest output of a function over a specified domain.

  • Minimum value: The smallest output of a function over a specified domain.

  • Extremum: A maximum or minimum.

  • Extrema: The plural of extremum.

  • Local maximum: A value greater than or equal to nearby function values.

  • Local minimum: A value less than or equal to nearby function values.

  • Absolute maximum: The greatest output over the entire specified domain.

  • Absolute minimum: The smallest output over the entire specified domain.

  • Turning point: A point where a graph changes from increasing to decreasing or vice versa.

  • Vertex: The turning point of a parabola.

  • Axis of symmetry: A line dividing a symmetric graph into matching halves.

  • Stationary point: A point where the derivative equals zero.

  • Derivative: A measure of the instantaneous rate of change of a function.

  • Critical point: An interior point where the derivative is zero or undefined, under the usual calculus definition.

  • Optimisation: The process of finding the greatest or smallest possible value of a quantity.

  • Restricted domain: A specified subset of permitted input values.

  • Endpoint: A boundary value of an interval.

  • Quadratic function: A function with a squared variable as its highest power.

  • Vertex form: A quadratic equation written as y = a(x − h)² + k.

  • Completing the square: An algebraic method for rewriting a quadratic expression in squared form.

  Key Takeaways

  • Maximums and minimums describe the greatest and smallest outputs of functions.

  • A maximum or minimum value is a y-value, while an extremum point includes both coordinates.

  • Local extrema describe behaviour near a point.

  • Absolute extrema describe the greatest or smallest values over the specified domain.

  • Turning points often occur where a graph changes between increasing and decreasing.

  • Upward-opening parabolas have a minimum at the vertex.

  • Downward-opening parabolas have a maximum at the vertex.

  • The vertex of y = ax² + bx + c occurs at x = −b/(2a).

  • Vertex form y = a(x − h)² + k makes the extremum easy to identify.

  • Restricted domains can change the absolute maximum and minimum.

  • Endpoints must be considered when finding absolute extrema on closed intervals.

  • Some functions have no absolute maximum or minimum.

  • Extrema are useful for solving real-world optimisation problems.

  • Comparing extrema helps us evaluate different functions, models, and situations.

   

  Check Your Understanding

  Recall and Understanding

  1. Explain the difference between a maximum value and a maximum point.

  2. What is the difference between a local maximum and an absolute maximum?

  3. Describe how to identify a minimum point from a graph.

  4. Explain why a linear function such as y = 3x + 2 has no absolute maximum or minimum over all real numbers.

  Application

  5. Find the minimum value and vertex of f(x) = x² − 8x + 10.

  6. Find the maximum value and vertex of g(x) = −2x² + 12x − 7.

  7. Identify the extremum of h(x) = 3(x − 4)² + 6.

  8. Identify the extremum of p(x) = −5(x + 2)² + 11.

  9. Find the absolute maximum and minimum of f(x) = x² − 4x + 1 on the interval 0 ≤ x ≤ 5.

  10. Determine the maximum and minimum values of y = 4 sin x − 3.

  Analysis and Challenge

  11. A ball's height is modeled by h(t) = −5t² + 40t + 3. Find its maximum height and the time at which it occurs.

  12. A company's profit is modeled by P(x) = −4x² + 160x − 900. Find the production quantity that maximises profit and calculate the maximum profit.

  13. A rectangle has a perimeter of 60 cm. Find the dimensions that maximise its area and calculate that area.

  14. Compare the extrema of:

  A: f(x) = −(x − 3)² + 12

  B: g(x) = −3(x − 3)² + 18

  C: h(x) = 2(x − 3)² − 5

  15. A student claims that the function f(x) = x³ − 3x has an absolute maximum at (−1, 2) because this point is a local maximum. Explain why the claim is incorrect.

  Answers to Check Your Understanding

  1. The maximum value is the greatest y-value of a function over a specified domain. The maximum point gives both the x-coordinate where it occurs and the maximum y-value.

  For example, if the maximum point is (4, 10), the maximum value is 10.

  2. A local maximum is greater than or equal to nearby outputs. An absolute maximum is the greatest output over the entire specified domain.

  3. Look for a point where the graph stops decreasing and begins increasing. This is a local minimum. To determine whether it is an absolute minimum, compare it with all other relevant values, including endpoints.

  4. The function y = 3x + 2 increases without bound as x increases and decreases without bound as x decreases. Therefore, it has neither an absolute maximum nor an absolute minimum over all real numbers.

  5.

  f(x) = x² − 8x + 10

  a = 1, b = −8

  x = −b/(2a)

  x = 8/2 = 4

  Substitute:

  f(4) = 16 − 32 + 10

  f(4) = −6

  Minimum value = −6

  Vertex = (4, −6)

  6.

  g(x) = −2x² + 12x − 7

  a = −2, b = 12

  x = −12/[2(−2)]

  x = 3

  Substitute:

  g(3) = −2(9) + 12(3) − 7

  g(3) = −18 + 36 − 7

  g(3) = 11

  Maximum value = 11

  Vertex = (3, 11)

  7.

  h(x) = 3(x − 4)² + 6

  The vertex is (4, 6).

  Because the quadratic coefficient is positive, the parabola opens upward.

  Minimum value = 6 at x = 4

  8.

  p(x) = −5(x + 2)² + 11

  The vertex is (−2, 11).

  Because the quadratic coefficient is negative, the parabola opens downward.

  Maximum value = 11 at x = −2

  9.

  f(x) = x² − 4x + 1

  Domain:

  0 ≤ x ≤ 5

  Find the vertex:

  x = −(−4)/(2 × 1) = 2

  Evaluate:

  f(0) = 1

  f(2) = 4 − 8 + 1 = −3

  f(5) = 25 − 20 + 1 = 6

  Compare:

  • f(0) = 1

  • f(2) = −3

  • f(5) = 6

  Absolute minimum = −3 at x = 2

  Absolute maximum = 6 at x = 5

  10.

  y = 4 sin x − 3

  Since:

  −1 ≤ sin x ≤ 1

  Multiplying by 4:

  −4 ≤ 4 sin x ≤ 4

  Subtracting 3:

  −7 ≤ 4 sin x − 3 ≤ 1

  Therefore:

  Maximum value = 1

  Minimum value = −7

  11.

  h(t) = −5t² + 40t + 3

  Find the vertex:

  t = −40/[2(−5)]

  t = 4

  Calculate:

  h(4) = −5(4)² + 40(4) + 3

  h(4) = −80 + 160 + 3

  h(4) = 83

  Maximum height = 83 m

  Time to maximum height = 4 s

  12.

  P(x) = −4x² + 160x − 900

  The vertex occurs at:

  x = −160/[2(−4)]

  x = 20

  Substitute:

  P(20) = −4(20)² + 160(20) − 900

  P(20) = −1600 + 3200 − 900

  P(20) = 700

  Optimal production quantity = 20 products

  Maximum modeled profit = $700

  13.

  Let the rectangle have length L and width W.

  Perimeter:

  2L + 2W = 60

  L + W = 30

  L = 30 − W

  Area:

  A = LW

  A(W) = W(30 − W)

  A(W) = −W² + 30W

  Find the vertex:

  W = −30/[2(−1)]

  W = 15

  Therefore:

  L = 30 − 15 = 15

  Area:

  A = 15 × 15 = 225

  Dimensions = 15 cm × 15 cm

  Maximum area = 225 cm²

  14.

  Function	Vertex	Extremum
  A	(3, 12)	Maximum of 12
  B	(3, 18)	Maximum of 18
  C	(3, −5)	Minimum of −5

  All three functions have their extrema at x = 3.

  Function B has the greatest maximum value.

  Function C opens upward and therefore has a minimum rather than a maximum.

  Function B is narrower than Function A because its quadratic coefficient has a greater absolute value.

  15.

  The function:

  f(x) = x³ − 3x

  has a local maximum at (−1, 2).

  However, it continues increasing without bound as x approaches positive infinity.

  Therefore, there are values of x for which f(x) is greater than 2.

  The point (−1, 2) is only a local maximum.

  The function has no absolute maximum over all real numbers.

   

  Final Summary

  Maximums and minimums are essential features of functions because they help identify the greatest and smallest possible values of quantities.

  By locating turning points, analysing quadratic equations, examining graphs, and considering restricted domains, we can determine where extrema occur and what they mean.

  These skills are particularly important in optimisation problems involving height, profit, cost, area, temperature, and other real-world quantities.

  The central idea is that finding an extremum requires understanding both the value of a function and the domain over which that value is being compared.

4. Function Features

Introduction

A graph provides much more information than simply showing how two variables are related. By examining its shape, position, intercepts, turning points, and overall direction, we can understand the behaviour of a function without calculating every possible output.

For example, consider the quadratic function:

f(x) = x² − 4

From its graph, we can determine that:

• It is symmetric about the y-axis.

• It has x-intercepts at (−2, 0) and (2, 0).

• It has a minimum value of −4 at x = 0.

• It is positive when x < −2 or x > 2.

• It is negative when −2 < x < 2.

• Both ends of the graph rise toward positive infinity.

These characteristics are called function features.

Understanding function features allows us to describe, compare, and interpret mathematical relationships in a systematic way.

This is useful in many areas of mathematics, including algebra, calculus, statistics, and mathematical modelling.

It is also important in science, engineering, economics, and other subjects where graphs are used to represent changing quantities.

What Are Function Features?

Function features are the important mathematical characteristics that describe a function's graph and behaviour.

Common features include:

• Domain and range.

• x-intercepts and y-intercepts.

• Symmetry.

• Increasing and decreasing intervals.

• Maximum and minimum values.

• Positive and negative intervals.

• End behaviour.

• Turning points.

• Asymptotes.

• Continuity.

Different function families have different combinations of these features.

For example, a straight line may have no turning points, while a quadratic function has one turning point.

A cubic function may have two turning points, and an exponential function may approach an asymptote without crossing it.

By identifying these features, we can build a complete mathematical description of a graph.

Symmetry of Functions

A graph has symmetry when parts of it match under a reflection or rotation.

Symmetry helps us understand graph shapes and predict values without calculating every point.

Two particularly important types of symmetry are:

• Symmetry about the y-axis.

• Symmetry about the origin.

These are associated with even functions and odd functions.

Even Functions: Symmetry About the y-Axis

An even function has a graph that is symmetric about the y-axis.

This means the left and right sides are mirror images.

The mathematical condition is:

f(−x) = f(x)

for every x in the domain.

The domain must also be symmetric about zero.

Example: A quadratic function

Consider:

f(x) = x²

Calculate:

f(2) = 2² = 4

f(−2) = (−2)² = 4

Therefore:

f(−2) = f(2)

The same relationship holds for every real x.

The function is even.

Other examples of even functions

• f(x) = x⁴

• f(x) = |x|

• f(x) = cos x

• f(x) = x² + 5

• f(x) = 3x⁴ − 2x² + 1

For example:

f(x) = 3x⁴ − 2x² + 1

Substitute −x:

f(−x) = 3(−x)⁴ − 2(−x)² + 1

Because even powers remove negative signs:

f(−x) = 3x⁴ − 2x² + 1

Therefore:

f(−x) = f(x)

The function is even.

Odd Functions: Symmetry About the Origin

An odd function has symmetry about the origin.

This means that rotating its graph 180° around the origin produces the same graph.

The mathematical condition is:

f(−x) = −f(x)

for every x in the domain.

Example: A cubic function

Consider:

f(x) = x³

Calculate:

f(2) = 8

f(−2) = −8

Therefore:

f(−2) = −f(2)

The same relationship holds for every real x.

The function is odd.

Other examples of odd functions

• f(x) = x

• f(x) = x⁵

• f(x) = sin x

• f(x) = 1/x, for x ≠ 0

• f(x) = x³ − 4x

Worked Example

Determine whether:

f(x) = x³ − 4x

is odd.

Substitute −x:

f(−x) = (−x)³ − 4(−x)

f(−x) = −x³ + 4x

Factor out −1:

f(−x) = −(x³ − 4x)

Therefore:

f(−x) = −f(x)

The function is odd.

Functions That Are Neither Even Nor Odd

Some functions have neither y-axis symmetry nor origin symmetry.

Consider:

f(x) = x² + x

Substitute −x:

f(−x) = x² − x

This is not equal to f(x).

It is also not equal to −f(x).

Therefore, the function is neither even nor odd.

Important distinction

A graph can have a line of symmetry without being an even function.

For example:

f(x) = (x − 3)²

has a vertical line of symmetry:

x = 3

However, it is not symmetric about the y-axis, so it is not an even function.

Similarly, some graphs have rotational symmetry around a point other than the origin without being odd functions.

Symmetry comparison

Feature	Even function	Odd function
Symmetry	y-axis	Origin
Algebraic test	f(−x) = f(x)	f(−x) = −f(x)
Typical example	x²	x³
Matching points	(x, y) and (−x, y)	(x, y) and (−x, −y)

The zero function f(x) = 0 is a special case: it is both even and odd.

Positive and Negative Intervals

Another important feature of a function is whether its outputs are positive or negative.

A function is positive wherever its graph lies above the x-axis.

A function is negative wherever its graph lies below the x-axis.

Mathematically:

• Positive: f(x) > 0

• Negative: f(x) < 0

• Zero: f(x) = 0

These intervals describe the sign of the function's outputs, not whether the graph is increasing or decreasing.

A function can be positive and decreasing at the same time.

Worked Example: A linear function

Consider:

f(x) = 2x − 6

First, find the x-intercept:

0 = 2x − 6

x = 3

The graph crosses the x-axis at (3, 0).

For x > 3:

2x − 6 > 0

Therefore, the function is positive.

For x < 3:

2x − 6 < 0

Therefore, the function is negative.

Positive interval: (3, ∞)

Negative interval: (−∞, 3)

At x = 3, the function equals zero, so this point belongs to neither interval.

Finding Positive and Negative Intervals of Quadratic Functions

Quadratic functions can have several sign patterns depending on their roots and the direction in which they open.

Worked Example: An upward-opening parabola

Consider:

f(x) = x² − 4

Factor:

f(x) = (x − 2)(x + 2)

The roots are:

x = −2 and x = 2

These divide the number line into three intervals:

• x < −2

• −2 < x < 2

• x > 2

Test one value in each interval.

Interval	Test x	f(x)	Sign
x < −2	−3	5	Positive
−2 < x < 2	0	−4	Negative
x > 2	3	5	Positive

Therefore:

Positive: (−∞, −2) and (2, ∞)

Negative: (−2, 2)

Worked Example: A downward-opening parabola

Consider:

g(x) = −x² + 9

Find the roots:

0 = −x² + 9

x² = 9

x = −3 or x = 3

Since the parabola opens downward, its graph lies above the x-axis between the roots and below the x-axis outside them.

Therefore:

Positive: (−3, 3)

Negative: (−∞, −3) and (3, ∞)

A parabola that never crosses the x-axis

Consider:

h(x) = x² + 2

Since x² ≥ 0:

x² + 2 ≥ 2

Therefore, the function is positive for every real x.

Positive: (−∞, ∞)

Negative: None

This demonstrates that a function does not need x-intercepts to have positive or negative intervals.

Sign Charts

A sign chart is a useful tool for identifying where a function is positive, negative, or zero.

Consider:

f(x) = (x + 3)(x − 1)

The roots are x = −3 and x = 1.

Interval	x + 3	x − 1	f(x)
x < −3	Negative	Negative	Positive
−3 < x < 1	Positive	Negative	Negative
x > 1	Positive	Positive	Positive

Multiplying two negative numbers gives a positive result.

Multiplying a positive number and a negative number gives a negative result.

Therefore:

Positive: (−∞, −3) and (1, ∞)

Negative: (−3, 1)

Sign charts are particularly useful for polynomial and rational functions.

For rational functions, remember to exclude any values where the denominator equals zero.

Distinguishing Sign from Direction of Change

One of the most important skills in graph analysis is distinguishing between the sign of a function and whether it is increasing or decreasing.

Consider:

f(x) = −x + 5

This function is decreasing everywhere because its gradient is −1.

However:

• It is positive for x < 5.

• It equals zero at x = 5.

• It is negative for x > 5.

The function is therefore positive and decreasing on (−∞, 5).

Similarly, the function:

g(x) = x − 5

is increasing everywhere but negative when x < 5.

Feature	Question being answered
Positive or negative	Is the graph above or below the x-axis?
Increasing or decreasing	Does the graph rise or fall from left to right?
Maximum or minimum	Where does the graph reach a high or low value?

These are different properties and should be analysed separately.

End Behaviour of Functions

End behaviour describes what happens to a function's outputs as x becomes very large and positive or very large and negative.

We examine two directions:

• As x approaches positive infinity.

• As x approaches negative infinity.

The symbol ∞ represents infinity.

The symbol −∞ represents negative infinity.

Infinity is not a number that a function eventually reaches. Instead, these expressions describe what happens as the input grows without bound.

End behaviour notation

We commonly write:

As x → ∞, f(x) → ∞

This means that as x becomes increasingly large and positive, the output increases without bound.

Similarly:

As x → −∞, f(x) → −∞

means that as x becomes increasingly large in the negative direction, the output decreases without bound.

End behaviour helps us understand the overall shape of a graph, especially outside the visible plotting window.

End Behaviour of Linear Functions

Consider:

f(x) = 2x + 3

The gradient is positive.

Therefore:

• As x → ∞, f(x) → ∞.

• As x → −∞, f(x) → −∞.

Now consider:

g(x) = −2x + 3

The gradient is negative.

Therefore:

• As x → ∞, g(x) → −∞.

• As x → −∞, g(x) → ∞.

The sign of the gradient determines the end behaviour of a nonconstant linear function.

End Behaviour of Quadratic Functions

For:

f(x) = ax² + bx + c

the sign of a determines the end behaviour.

Positive leading coefficient

If a > 0, the parabola opens upward.

Both ends rise:

As x → ∞, f(x) → ∞

As x → −∞, f(x) → ∞

Example:

f(x) = x² − 4x + 1

Negative leading coefficient

If a < 0, the parabola opens downward.

Both ends fall:

As x → ∞, f(x) → −∞

As x → −∞, f(x) → −∞

Example:

g(x) = −2x² + 3x + 5

The lower-degree terms affect the position and shape of the graph but do not change its end behaviour.

End Behaviour of Polynomial Functions

For polynomial functions, the highest-power term usually determines the end behaviour.

This term is called the leading term.

For example:

f(x) = 3x⁴ − 2x³ + 5x − 7

The leading term is:

3x⁴

Because the degree is even and the leading coefficient is positive, both ends rise.

Four common end behaviour patterns

Summary table

Degree	Leading coefficient	Left end	Right end
Even	Positive	Rises	Rises
Even	Negative	Falls	Falls
Odd	Positive	Falls	Rises
Odd	Negative	Rises	Falls

Worked Example

Describe the end behaviour of:

f(x) = −4x⁵ + 3x² − 8

The leading term is:

−4x⁵

The degree is odd.

The leading coefficient is negative.

Therefore:

As x → −∞, f(x) → ∞

As x → ∞, f(x) → −∞

The graph rises on the left and falls on the right.

End Behaviour of Exponential Functions

Exponential functions behave differently from polynomials.

Consider:

f(x) = 2ˣ

As x becomes increasingly large:

f(x) increases without bound.

Therefore:

As x → ∞, f(x) → ∞

As x becomes increasingly negative:

f(x) approaches zero from above.

Therefore:

As x → −∞, f(x) → 0

The line y = 0 is a horizontal asymptote.

Exponential decay

Consider:

g(x) = (1/2)ˣ

As x → ∞:

g(x) → 0

As x → −∞:

g(x) → ∞

The function decreases from left to right and approaches zero on the right.

Vertical translations and asymptotes

Consider:

h(x) = 2ˣ + 3

This is the graph of 2ˣ translated upward by 3 units.

Its horizontal asymptote is:

y = 3

Therefore:

As x → −∞, h(x) → 3.

As x → ∞, h(x) → ∞.

The vertical translation changes the horizontal asymptote but does not change the overall direction of exponential growth.

File:Exponential function y = 2^x.svg - Wikimedia Commons

 

 

 

End Behaviour of Rational Functions

Rational functions are ratios of polynomial expressions.

Consider:

f(x) = 1/x

As x becomes very large and positive:

f(x) approaches zero from above.

As x becomes very large and negative:

f(x) approaches zero from below.

Therefore:

• As x → ∞, f(x) → 0.

• As x → −∞, f(x) → 0.

The graph has a horizontal asymptote:

y = 0

It also has a vertical asymptote at:

x = 0

The function is undefined at x = 0.

Worked Example

Consider:

g(x) = 3 + 2/x

As x → ∞:

2/x → 0

Therefore:

g(x) → 3

As x → −∞:

2/x → 0

Therefore:

g(x) → 3

The horizontal asymptote is:

y = 3

This illustrates how a function can approach a finite value at both ends rather than increasing or decreasing without bound.

End Behaviour of Trigonometric Functions

Periodic functions such as sine and cosine behave differently again.

Consider:

f(x) = sin x

The function repeatedly oscillates between −1 and 1.

As x approaches positive or negative infinity, the function continues oscillating.

It does not approach one particular value.

Therefore, sin x has no single limiting value as x → ∞ or x → −∞.

This demonstrates that not all functions have end behaviour described by approaching infinity or a horizontal asymptote.

Comparing Important Graph Features

When comparing functions, it is useful to organise the information into a table.

Consider these four functions:

A: f(x) = x²

B: g(x) = −x²

C: h(x) = x³

D: p(x) = 2ˣ

Comparison table

Feature	x²	−x²	x³	2ˣ
Domain	All real numbers	All real numbers	All real numbers	All real numbers
Range	y ≥ 0	y ≤ 0	All real numbers	y > 0
Symmetry	Even	Even	Odd	Neither
x-intercept	(0, 0)	(0, 0)	(0, 0)	None
y-intercept	(0, 0)	(0, 0)	(0, 0)	(0, 1)
Extremum	Minimum 0	Maximum 0	None	None
Positive intervals	x ≠ 0	None	x > 0	All real x
Negative intervals	None	x ≠ 0	x < 0	None

Comparing their end behaviour

f(x) = x²

Both ends rise.

g(x) = −x²

Both ends fall.

h(x) = x³

The left end falls and the right end rises.

p(x) = 2ˣ

The left end approaches zero, while the right end rises without bound.

Why comparisons matter

Although these functions may pass through some of the same points, their overall behaviour is very different.

For example, x² and x³ both pass through the origin, but:

• x² is even.

• x³ is odd.

• x² has a minimum.

• x³ has no local or absolute extrema over all real numbers.

• x² is never negative.

• x³ is negative for negative x-values.

Comparing several features gives a more complete understanding than looking at intercepts alone.

Summarising the Behaviour of a Graph

A complete graph description should combine several important features.

Rather than listing disconnected facts, we can explain how the function behaves as a whole.

Worked Example: A quadratic function

Consider:

f(x) = x² − 6x + 5

We will analyse its important features.

Finding the intercepts

For the x-intercepts, set f(x) = 0:

x² − 6x + 5 = 0

Factor:

(x − 1)(x − 5) = 0

Therefore:

x = 1 or x = 5

x-intercepts: (1, 0) and (5, 0)

For the y-intercept:

f(0) = 5

y-intercept: (0, 5)

Finding the turning point

Use:

x = −b/(2a)

Here a = 1 and b = −6.

x = 6/2 = 3

Substitute:

f(3) = 9 − 18 + 5

f(3) = −4

Vertex: (3, −4)

Since the parabola opens upward, this is a minimum.

Finding the symmetry

The axis of symmetry is:

x = 3

The graph is symmetric about this vertical line.

However, the function is neither even nor odd because it is not symmetric about the y-axis or origin.

Finding the positive and negative intervals

The roots are x = 1 and x = 5.

The parabola opens upward.

Therefore:

Positive: (−∞, 1) and (5, ∞)

Negative: (1, 5)

Finding increasing and decreasing intervals

The vertex occurs at x = 3.

Therefore:

Decreasing: (−∞, 3)

Increasing: (3, ∞)

Describing end behaviour

The leading term is x².

It has even degree and a positive leading coefficient.

Therefore:

• As x → −∞, f(x) → ∞.

• As x → ∞, f(x) → ∞.

Complete graph summary

Feature	Description
Function	f(x) = x² − 6x + 5
Domain	All real numbers
Range	y ≥ −4
x-intercepts	(1, 0), (5, 0)
y-intercept	(0, 5)
Minimum	−4 at x = 3
Axis of symmetry	x = 3
Even or odd	Neither
Increasing	(3, ∞)
Decreasing	(−∞, 3)
Positive	(−∞, 1), (5, ∞)
Negative	(1, 5)
End behaviour	Both ends rise

This is a complete mathematical description of the function's main features.

Real-World Application: Projectile Motion

A ball is thrown upward from a platform.

Its height is modeled by:

h(t) = −5t² + 20t + 5

where:

• t is time in seconds.

• h is height in metres.

Initial height

At t = 0:

h(0) = 5

The ball begins 5 m above the ground.

Maximum height

The vertex occurs at:

t = −20/[2(−5)]

t = 2

Calculate:

h(2) = −5(2)² + 20(2) + 5

h(2) = −20 + 40 + 5

h(2) = 25

Maximum height = 25 m at t = 2 s

When the ball reaches the ground

Set h(t) = 0:

−5t² + 20t + 5 = 0

Divide by −5:

t² − 4t − 1 = 0

Using the quadratic formula:

t = 2 ± √5

The positive solution is:

t ≈ 4.24 s

Interpreting the function features

Over the physical flight:

• Height increases from t = 0 to t = 2 s.

• Maximum height is 25 m.

• Height decreases from t = 2 s to approximately t = 4.24 s.

• Height is positive before the ball lands.

• Height equals zero when the ball reaches the ground.

Although the mathematical quadratic has end behaviour extending to infinity, this is not physically meaningful for the ball's actual flight.

The practical domain is approximately:

0 ≤ t ≤ 4.24

This illustrates why real-world models must be interpreted within appropriate domains.

Soccer ball

 

 

 

Real-World Application: Business Profit

Suppose a business models its profit using:

P(x) = −2(x − 30)² + 1800

where:

• x is the number of products sold.

• P(x) is profit in dollars.

Maximum profit

The vertex is:

(30, 1800)

Therefore:

Maximum modeled profit = $1,800

This occurs when 30 products are sold.

Positive and negative profit

A positive profit means the business earns more revenue than its costs.

A negative profit means the business operates at a loss.

Find the break-even points by setting P(x) = 0:

0 = −2(x − 30)² + 1800

2(x − 30)² = 1800

(x − 30)² = 900

x − 30 = ±30

x = 0 or x = 60

Therefore:

• Profit is positive for 0 < x < 60.

• Profit equals zero at x = 0 and x = 60.

• The mathematical model gives negative profit for x > 60.

Interpreting trends

Profit increases until 30 products are sold and decreases afterward.

The function is symmetric about:

x = 30

This means equal deviations above and below 30 produce equal modeled profits.

For example:

P(20) = −2(20 − 30)² + 1800 = 1600

P(40) = −2(40 − 30)² + 1800 = 1600

Both production quantities produce a modeled profit of $1,600.

The model is illustrative and assumes the quadratic relationship remains appropriate over the quantities being considered.

Real-World Application: Population Growth

Consider:

P(t) = 500(1.08)ᵗ

where t is time in years.

The population starts at 500 and increases by 8% each year.

Function features

• y-intercept: (0, 500).

• Increasing for all real t mathematically.

• Positive for all real t.

• No x-intercept.

• No maximum or minimum over all real t.

• Horizontal asymptote: P = 0.

• As t → ∞, P(t) → ∞.

• As t → −∞, P(t) → 0.

For a real population model, we might restrict the domain to t ≥ 0.

Over this practical domain, the minimum modeled population is 500 at t = 0.

This is another example of how restricting the domain changes the extrema of a function.

This consumer microbiome startup is betting profits on an AI tool

 

 

 

A Systematic Method for Analysing Any Graph

When given a graph or equation, use the following process.

Identify the function family.

Determine whether the graph is linear, quadratic, cubic, exponential, rational, trigonometric, or another type.

Find the domain and range.

Determine the allowed input values and possible outputs.

Find the intercepts.

Calculate or estimate where the graph meets the coordinate axes.

Investigate symmetry.

Check whether the graph has y-axis symmetry, origin symmetry, or another line of symmetry.

Identify positive and negative intervals.

Find where the graph lies above and below the x-axis.

Identify increasing and decreasing intervals.

Read the graph from left to right and locate changes in direction.

Locate extrema.

Identify local and absolute maximums and minimums where they exist.

Describe end behaviour.

Explain what happens as x approaches positive and negative infinity, when this is appropriate for the domain.

Identify asymptotes or discontinuities.

Check whether the graph approaches particular lines or contains gaps.

Summarise the overall behaviour.

Combine the important features into a clear mathematical description.

Common Mistakes

Confusing symmetry about the y-axis with symmetry about another vertical line

A parabola with axis x = 3 is not necessarily an even function.

Assuming every function is even or odd

Many functions are neither.

Confusing positive intervals with increasing intervals

Positive means the output is above zero. Increasing means the output rises as x increases.

Including roots in strictly positive or negative intervals

At an x-intercept, f(x) = 0, so the function is neither positive nor negative.

Describing end behaviour using only the visible graph

A plotting window may not show the long-term behaviour of a function.

Ignoring the leading coefficient

For polynomial functions, the sign of the leading coefficient affects the direction of the graph's ends.

Assuming all functions approach infinity at their ends

Exponential and rational functions may approach finite asymptotes, while trigonometric functions may oscillate indefinitely.

Confusing an asymptote with an intercept

An asymptote describes limiting behaviour, while an intercept is a point on the graph.

Assuming symmetry guarantees a maximum or minimum

For example, y = x³ has origin symmetry but no maximum or minimum over all real numbers.

Ignoring domain restrictions

A real-world model may only apply over a limited interval, changing which features are relevant.

Key Terms

• Function feature: A mathematical characteristic describing a graph or function.

• Symmetry: A property in which a graph matches itself under a reflection or rotation.

• Even function: A function satisfying f(−x) = f(x).

• Odd function: A function satisfying f(−x) = −f(x).

• Axis of symmetry: A line dividing a graph into mirror-image halves.

• Origin symmetry: Symmetry under a 180° rotation about the origin.

• Positive interval: An interval where f(x) > 0.

• Negative interval: An interval where f(x) < 0.

• Root or zero: An input value where f(x) = 0.

• Sign chart: A table showing where an expression is positive or negative.

• End behaviour: The behaviour of a function as its input approaches positive or negative infinity.

• Leading term: The term containing the highest power in a polynomial.

• Leading coefficient: The coefficient of the highest-power term.

• Degree: The highest exponent of the variable in a polynomial.

• Asymptote: A line that a graph approaches in a specified limiting direction.

• Horizontal asymptote: A horizontal line approached by a graph as x approaches positive or negative infinity.

• Vertical asymptote: A vertical line near which a function becomes unbounded on at least one side.

• Increasing interval: An interval where outputs rise as inputs increase.

• Decreasing interval: An interval where outputs fall as inputs increase.

• Extremum: A maximum or minimum value.

• Turning point: A point where a graph changes from increasing to decreasing or vice versa.

• Domain: The set of allowed input values.

• Range: The set of possible output values.

Key Takeaways

• Function features provide a systematic way to describe and compare graphs.

• Even functions have y-axis symmetry and satisfy f(−x) = f(x).

• Odd functions have origin symmetry and satisfy f(−x) = −f(x).

• A function may be neither even nor odd.

• Positive intervals occur where the graph lies above the x-axis.

• Negative intervals occur where the graph lies below the x-axis.

• Positive and negative intervals are different from increasing and decreasing intervals.

• End behaviour describes how a function behaves as x approaches positive or negative infinity.

• The leading term determines the end behaviour of a nonconstant polynomial.

• Exponential and rational functions may approach horizontal asymptotes.

• Periodic functions may continue oscillating rather than approaching a single value.

• Graphs can be compared using intercepts, symmetry, extrema, intervals, and end behaviour.

• Real-world graph analysis must consider the practical domain and meaning of the variables.

 

Check Your Understanding

Recall and Understanding

1. What is an even function? State the algebraic condition used to identify one.

2. What is an odd function? Describe its graphical symmetry.

3. Explain the difference between a positive interval and an increasing interval.

4. What is meant by the end behaviour of a function?

5. Why is the leading term important when describing the end behaviour of a polynomial?

Application

6. Determine whether f(x) = 4x⁴ − 3x² + 2 is even, odd, or neither.

7. Determine whether g(x) = 2x³ − 5x is even, odd, or neither.

8. Find the positive and negative intervals of f(x) = x² − 9.

9. Find the positive and negative intervals of g(x) = −x² + 16.

10. Describe the end behaviour of f(x) = 3x⁴ − 2x + 7.

11. Describe the end behaviour of g(x) = −2x⁵ + x³ − 4.

12. Determine the horizontal asymptote and end behaviour of h(x) = 3ˣ + 2.

Analysis and Challenge

13. Analyse f(x) = x² − 4x + 3. Identify its intercepts, axis of symmetry, minimum, positive and negative intervals, increasing and decreasing intervals, and end behaviour.

14. Compare the symmetry, extrema, and end behaviour of f(x) = x² and g(x) = −x³.

15. A student claims that f(x) = (x − 2)² is even because its graph has a vertical axis of symmetry. Explain the error.

Answers to Check Your Understanding

1. An even function has symmetry about the y-axis and satisfies:

f(−x) = f(x)

for every input in its domain.

2. An odd function has symmetry about the origin. Rotating its graph 180° around the origin produces the same graph.

Its algebraic condition is:

f(−x) = −f(x)

3. A positive interval is where f(x) > 0, meaning the graph is above the x-axis. An increasing interval is where the output rises as x increases.

4. End behaviour describes what happens to the outputs as x approaches positive or negative infinity.

5. For a polynomial, the leading term dominates the lower-degree terms when the magnitude of x becomes very large. Its degree and coefficient determine the graph's end behaviour.

6.

f(x) = 4x⁴ − 3x² + 2

f(−x) = 4(−x)⁴ − 3(−x)² + 2

f(−x) = 4x⁴ − 3x² + 2

Therefore:

The function is even.

7.

g(x) = 2x³ − 5x

g(−x) = −2x³ + 5x

g(−x) = −g(x)

Therefore:

The function is odd.

8.

f(x) = x² − 9

Factor:

f(x) = (x − 3)(x + 3)

Roots:

x = −3 and x = 3

Since the parabola opens upward:

Positive: (−∞, −3) and (3, ∞)

Negative: (−3, 3)

9.

g(x) = −x² + 16

Set g(x) = 0:

x² = 16

x = −4 or x = 4

The parabola opens downward.

Positive: (−4, 4)

Negative: (−∞, −4) and (4, ∞)

10.

f(x) = 3x⁴ − 2x + 7

The leading term is 3x⁴.

It has even degree and a positive leading coefficient.

Therefore:

As x → −∞, f(x) → ∞

As x → ∞, f(x) → ∞

Both ends rise.

11.

g(x) = −2x⁵ + x³ − 4

The leading term is −2x⁵.

It has odd degree and a negative leading coefficient.

Therefore:

As x → −∞, g(x) → ∞

As x → ∞, g(x) → −∞

The left end rises and the right end falls.

12.

h(x) = 3ˣ + 2

The parent exponential function is shifted upward by 2.

Horizontal asymptote:

y = 2

End behaviour:

As x → −∞, h(x) → 2.

As x → ∞, h(x) → ∞.

13.

f(x) = x² − 4x + 3

Factor:

f(x) = (x − 1)(x − 3)

x-intercepts:

(1, 0) and (3, 0)

y-intercept:

f(0) = 3

(0, 3)

Vertex x-coordinate:

x = −(−4)/(2 × 1) = 2

Vertex y-coordinate:

f(2) = 4 − 8 + 3 = −1

Minimum point:

(2, −1)

Axis of symmetry:

x = 2

Positive intervals:

(−∞, 1) and (3, ∞)

Negative interval:

(1, 3)

Decreasing interval:

(−∞, 2)

Increasing interval:

(2, ∞)

End behaviour:

Both ends rise toward positive infinity.

The function is neither even nor odd.

14.

For f(x) = x²:

• Even function.

• Symmetric about the y-axis.

• Minimum value 0 at x = 0.

• Both ends rise toward positive infinity.

For g(x) = −x³:

• Odd function.

• Symmetric about the origin.

• No local or absolute extrema over all real numbers.

• Left end rises toward positive infinity.

• Right end falls toward negative infinity.

Both functions pass through the origin, but their symmetry and end behaviour differ.

15.

The function:

f(x) = (x − 2)²

has a vertical axis of symmetry at:

x = 2

However, an even function must be symmetric specifically about the y-axis.

Check algebraically:

f(−x) = (−x − 2)² = (x + 2)²

This is not generally equal to (x − 2)².

Therefore:

The function is neither even nor odd.

 

Final Summary

Function features allow us to understand the overall behaviour of graphs by examining symmetry, positive and negative intervals, end behaviour, intercepts, extrema, and changes in direction.

By combining these characteristics, we can describe functions accurately, compare different mathematical relationships, and interpret graphs in real-world contexts.

A complete understanding of a graph comes from examining how its important features work together, rather than studying each feature separately.

 

5. Function Modeling Project

Introduction

Mathematics is not only about solving equations and drawing graphs. It is also a powerful tool for understanding real-world situations, identifying patterns, making predictions, and solving practical problems.

For example, mathematical functions can help us investigate questions such as:

• How does the height of a bouncing ball change over time?

• How quickly does a plant grow?

• How does the cost of a taxi journey depend on the distance travelled?

• How does the temperature of a hot drink change as it cools?

• How does the area of a rectangle change when its dimensions change?

• How can we predict future population growth?

These questions can be investigated through mathematical modelling.

Mathematical modelling is the process of representing a real-world situation using mathematical ideas such as equations, functions, tables, and graphs.

A mathematical model does not need to describe reality perfectly. Instead, it should capture the important relationships in a situation accurately enough to explain observations, make useful predictions, or support decisions.

In this project, we will explore how to collect data, identify patterns, select suitable functions, construct mathematical models, interpret results, and communicate conclusions.

What Is Mathematical Modelling?

A mathematical model is a simplified mathematical representation of a real-world situation.

Models may include:

• Equations.

• Functions.

• Tables of values.

• Graphs.

• Diagrams.

• Statistical relationships.

For example, suppose a bicycle rental company charges a fixed fee of $5 plus $3 per hour.

The total cost can be represented by:

C = 3h + 5

where:

• C is the total cost in dollars.

• h is the number of hours rented.

This equation is a mathematical model.

It allows us to calculate the rental cost for different durations.

For example, for 4 hours:

C(4) = 3(4) + 5

C(4) = 17

The total cost is $17.

The model also helps us understand the relationship:

• The cost increases by $3 for every additional hour.

• The initial fee is $5.

• The relationship is linear.

• The graph is a straight line.

A useful mathematical model should explain the observed relationship and make predictions that are reasonable within the situation being studied.

The Mathematical Modelling Cycle

Mathematical modelling is a process that often involves several stages.

A typical modelling project follows these stages:

1. Identify a real-world question.

2. Decide which variables to investigate.

3. Collect reliable data.

4. Organise the data in tables and graphs.

5. Identify patterns.

6. Select an appropriate function.

7. Construct an equation.

8. Compare the model with the observed data.

9. Interpret results and make predictions.

10. Communicate findings and evaluate limitations.

The process is not always completed in a straight line.

For example, after plotting the data, we may realise that the original measurements were insufficient or that another function would describe the situation more accurately.

Revising a model is a normal and important part of mathematical investigation.

Choosing a Real-World Situation

The first stage of a modelling project is selecting a situation that can be investigated mathematically.

A suitable project should involve two quantities that can be measured or calculated.

Examples include:

The choice of function should depend on the observed relationship, not simply on the topic.

For example, plant growth may appear approximately linear over a short period but may slow down as the plant approaches its mature size.

Identifying Variables

A mathematical model usually describes how one variable depends on another.

Independent variable

The independent variable is the input.

It is often represented by x.

Examples include:

• Time.

• Distance travelled.

• Number of products sold.

• Side length.

• Temperature.

Dependent variable

The dependent variable is the output.

It is often represented by y or f(x).

Examples include:

• Total cost.

• Height.

• Profit.

• Area.

• Population.

Worked Example

Suppose we investigate how the distance travelled by a toy car changes over time.

The independent variable is:

Time (seconds)

The dependent variable is:

Distance travelled (metres)

We can write:

d = f(t)

This means that distance is modelled as a function of time.

Controlled conditions

In practical investigations, other variables may influence the results.

For example, when measuring toy car motion, we should consider:

• Whether the car moves on the same surface.

• Whether the starting position remains the same.

• Whether the car travels along a straight path.

• Whether the method of measuring time is consistent.

Controlling these conditions helps make the data more reliable.

Collecting Real-World Data

Reliable data is essential for developing a useful mathematical model.

Data can be collected through:

• Direct measurements.

• Scientific experiments.

• Surveys.

• Observations.

• Public datasets.

• Digital sensors.

• Simulations.

Example: Measuring the motion of a toy car

Suppose a toy car travels along a straight track.

We record its distance from the starting point at regular time intervals.

The following is an illustrative dataset, not actual experimental measurements.

The data suggests that distance increases approximately linearly with time.

However, the measurements are not perfectly aligned.

This could be caused by:

• Measurement uncertainty.

• Small changes in speed.

• Human reaction time.

• Surface friction.

• Differences in how the car was released.

Real-world data often contains variation.

A mathematical model should describe the overall trend rather than necessarily passing through every measurement.

Organising Data in Tables

A data table should be clear and easy to interpret.

A good table includes:

• Descriptive headings.

• Correct units.

• Consistent numerical precision.

• Measurements arranged logically.

• Enough observations to identify patterns.

For example:

Avoid using headings such as simply "x" and "y" in a final scientific report when meaningful variable names are available.

Instead, write:

Time (s) and Distance (m)

This makes the data understandable to someone who did not conduct the experiment.

Creating Scatter Plots

A scatter plot represents pairs of measured values on a coordinate plane.

Each point represents one observation.

For the toy car experiment:

• Time is placed on the horizontal axis.

• Distance is placed on the vertical axis.

  

  The plotted points show an upward trend.

  This suggests that distance increases as time increases.

  The points are also approximately arranged along a straight line.

  Therefore, a linear function may be a suitable model.

  Important distinction

  A scatter plot shows the observed measurements.

  A model graph represents the mathematical relationship used to describe those measurements.

  The two should be compared, but they are not necessarily identical.

  Recognising Function Patterns

  One of the most important modelling skills is recognising the shape of a relationship.

  Different function families produce different patterns.

  Linear relationships

  A linear function has the form:

  y = mx + c

  Its graph is a straight line.

  A linear model is appropriate when the dependent variable changes by approximately the same amount for equal increases in the independent variable.

  Examples include:

  • Distance travelled at constant speed.

  • Cost with a fixed fee and constant rate.

  • Earnings at a fixed hourly wage.

  Quadratic relationships

  A quadratic function has the form:

  y = ax² + bx + c

  Its graph is a parabola.

  Quadratic models are useful when the rate of change changes approximately linearly with the input.

  Examples include:

  • Height of a projectile under constant gravitational acceleration.

  • Area as a function of side length.

  • Some simplified profit and revenue models.

  Exponential relationships

  An exponential function has the form:

  y = abˣ

  where a ≠ 0, b > 0, and b ≠ 1.

  Exponential models are useful when a quantity changes by approximately the same percentage over equal input intervals.

  Examples include:

  • Compound interest.

  • Idealised population growth.

  • Radioactive decay.

  • Cooling toward a constant surrounding temperature, using a shifted exponential model.

  Inverse relationships

  An inverse function may have the form:

  y = k/x

  where x ≠ 0.

  Examples include:

  • Time required to travel a fixed distance at different speeds.

  • Pressure and volume of an ideal gas at constant temperature, under suitable conditions.

  Periodic relationships

  Periodic functions repeat patterns over regular intervals.

  Sine and cosine functions are commonly used:

  y = A sin(B(x − h)) + k

  Examples include:

  • Oscillations.

  • Seasonal patterns.

  • Tides.

  • Rotating machinery.

  Comparing Function Families

  

  These graph shapes help us select possible models, but visual appearance alone is not enough.

  We must also consider the context, mathematical assumptions, and how well the model matches the data.

  Creating a Linear Model from Data

  Return to the toy car measurements.

  Suppose we estimate a line using two representative points:

  (0, 0) and (6, 9.1)

  The gradient is:

  m = (y₂ − y₁)/(x₂ − x₁)

  Substitute:

  m = (9.1 − 0)/(6 − 0)

  m ≈ 1.52

  The y-intercept is approximately zero.

  Therefore, a simple linear model is:

  d(t) = 1.52t

  where:

  • d is distance in metres.

  • t is time in seconds.

  Interpreting the gradient

  The gradient represents speed.

  A gradient of 1.52 means the toy car travels approximately 1.52 metres per second under this model.

  Making a prediction

  How far will the car travel after 8 seconds?

  d(8) = 1.52(8)

  d(8) = 12.16

  Predicted distance ≈ 12.16 m

  This prediction assumes the car continues moving at approximately the same speed.

  Evaluating the model

  The model predicts:

  Time (s)	Measured distance (m)	Predicted distance (m)
  0	0.0	0.00
  1	1.5	1.52
  2	3.1	3.04
  3	4.4	4.56
  4	6.2	6.08
  5	7.6	7.60
  6	9.1	9.12

  The predictions are reasonably close to the illustrative measurements.

  This supports using a linear model over the observed interval.

  However, the line was estimated from selected points rather than calculated using formal regression.

  A least-squares regression line could provide a more systematic fit.

  Residuals and Model Accuracy

  A residual is the difference between an observed value and the value predicted by a model.

  Residual = Observed value − Predicted value

  Worked Example

  At t = 3 seconds:

  Observed distance = 4.4 m

  Predicted distance = 1.52(3) = 4.56 m

  Residual:

  4.4 − 4.56 = −0.16 m

  The negative residual means the model overestimated the observed distance by 0.16 m.

  Interpreting residuals

  • Positive residual: the observed value is above the prediction.

  • Negative residual: the observed value is below the prediction.

  • Zero residual: the prediction matches the observation.

  A good model generally has relatively small residuals with no strong systematic pattern.

  If residuals consistently form a curve, a different function family may be more suitable.

  A model with very small residuals is not automatically the best model, however. It must also make sense in the real-world situation and avoid unnecessary complexity.

  Creating a Quadratic Model

  Suppose we investigate the height of a ball thrown vertically upward.

  Under a simplified model with constant gravitational acceleration and negligible air resistance:

  h(t) = −4.9t² + v₀t + h₀

  where:

  • h(t) = height in metres.

  • t = time in seconds.

  • v₀ = initial vertical velocity in metres per second.

  • h₀ = initial height in metres.

  Suppose the ball is released from a height of 1.5 m with an initial upward velocity of 14.7 m/s.

  Then:

  h(t) = −4.9t² + 14.7t + 1.5

  Finding the maximum height

  The vertex occurs at:

  t = −b/(2a)

  Here:

  a = −4.9

  b = 14.7

  Therefore:

  t = −14.7/[2(−4.9)]

  t = 1.5 s

  Substitute:

  h(1.5) = −4.9(1.5)² + 14.7(1.5) + 1.5

  h(1.5) = −11.025 + 22.05 + 1.5

  h(1.5) = 12.525

  Maximum height ≈ 12.53 m

  The ball reaches its maximum height after 1.5 seconds.

  Finding when the ball reaches the ground

  Set h(t) = 0:

  0 = −4.9t² + 14.7t + 1.5

  Using the quadratic formula, the positive solution is approximately:

  t ≈ 3.10 s

  The ball reaches the ground approximately 3.10 seconds after release.

  Interpreting the model

  The model predicts that:

  • The ball begins 1.5 m above the ground.

  • It rises until t = 1.5 s.

  • Its maximum height is approximately 12.53 m.

  • It then falls.

  • It reaches the ground at approximately 3.10 s.

  The practical domain is approximately:

  0 ≤ t ≤ 3.10

  The model should not be used to predict the ball's height long after it hits the ground.

  PPT - Speed, velocity and acceleration PowerPoint Presentation, free download - ID:3124611

   

   

   

  Creating an Exponential Model

  Suppose a population of microorganisms starts at 200 and increases by 15% every hour.

  The population can be modeled by:

  P(t) = 200(1.15)ᵗ

  where t is time in hours.

  Understanding the equation

  • 200 is the initial population.

  • 1.15 is the growth factor.

  • 0.15 represents a 15% increase per hour.

  • t is the number of hours.

  Making predictions

  After 4 hours:

  P(4) = 200(1.15)⁴

  P(4) ≈ 349.8

  The model predicts approximately 350 microorganisms.

  After 8 hours:

  P(8) = 200(1.15)⁸

  P(8) ≈ 611.8

  The model predicts approximately 612 microorganisms.

  Evaluating the model

  Exponential growth may describe a population over a limited period.

  However, real populations cannot increase indefinitely without constraints.

  Factors such as food supply, space, competition, and waste accumulation may eventually slow growth.

  Therefore, the model may be useful for short-term predictions but unsuitable for very long-term predictions.

  Microbiology laboratory at SINTEF SeaLab - SINTEF

   

   

   

  Interpolation and Extrapolation

  Models can be used to estimate values that were not directly measured.

  Two important terms are interpolation and extrapolation.

  Interpolation

  Interpolation means estimating a value within the range of collected data.

  For example, suppose we measured a car's distance between 0 and 6 seconds.

  Using the model to estimate distance at 2.5 seconds is interpolation.

  d(2.5) = 1.52(2.5)

  d(2.5) = 3.8 m

  Extrapolation

  Extrapolation means predicting beyond the range of collected data.

  Using the same model to estimate distance after 20 seconds is extrapolation.

  d(20) = 1.52(20)

  d(20) = 30.4 m

  However, this prediction may be unreliable if the car slows down, changes direction, or reaches the end of the track.

  Extrapolation generally requires greater caution because the relationship may change beyond the observed data.

  Using Technology to Build Models

  Digital tools make mathematical modelling easier.

  Common options include:

  • Desmos — plotting functions and using regression tools.

  • GeoGebra — graphing, regression, and mathematical exploration.

  • Microsoft Excel — organising data, scatter plots, and trendlines.

  • Google Sheets — tables, charts, and regression calculations.

  Example workflow using a spreadsheet

  1. Enter the independent variable in the first column.

  2. Enter the dependent variable in the second column.

  3. Highlight the data.

  4. Insert a scatter plot.

  5. Add a trendline.

  6. Choose a suitable model type.

  7. Display the trendline equation when supported.

  8. Compare the predicted values with the measurements.

  9. Use the equation to make predictions.

  Technology helps us fit models more efficiently, but we must still decide whether the model makes mathematical and practical sense.

  Evaluating a Mathematical Model

  A mathematical model should be evaluated rather than accepted automatically.

  Useful evaluation questions include:

  • Does the model describe the observed data accurately?

  • Does the equation have a reasonable interpretation?

  • Are the units correct?

  • Does the model follow the expected scientific or mathematical relationship?

  • Are there large differences between observed and predicted values?

  • Is the model suitable for predictions outside the observed interval?

  • What assumptions were made?

  • What factors were ignored?

  • Could a different function provide a better model?

  Accuracy versus usefulness

  A model does not have to match every observation perfectly to be useful.

  For example, a linear model may describe the motion of a toy car well enough to estimate its average speed, even if small variations occur in the measurements.

  However, if the car accelerates significantly, a quadratic model may provide a better description of distance over time.

  Model limitations

  All mathematical models simplify reality.

  Possible limitations include:

  • Measurement uncertainty.

  • Small sample sizes.

  • Uncontrolled variables.

  • Changes in environmental conditions.

  • Incorrect assumptions.

  • Limited data ranges.

  • Natural variation.

  • Instrument precision.

  A strong modelling report explains these limitations rather than ignoring them.

  Communicating Mathematical Findings

  Mathematical modelling is not complete until the results are clearly communicated.

  A good report should explain:

  • The question being investigated.

  • The data collection method.

  • The variables and their units.

  • The mathematical function selected.

  • The equation of the model.

  • The graph and important features.

  • The interpretation of coefficients.

  • Predictions made using the model.

  • Evidence supporting the model.

  • Limitations and possible improvements.

  Example of mathematical communication

  Suppose a student models bicycle rental cost using:

  C = 3h + 5

  A weak explanation might be:

  "The graph goes up and the equation works."

  A stronger explanation would be:

  "The relationship between rental duration and total cost is linear. The gradient of 3 represents the additional cost of $3 per hour, while the y-intercept of 5 represents the fixed rental fee. The model predicts a total cost of $17 for a four-hour rental. This model assumes that the hourly rate remains constant and that no additional charges apply."

  The stronger explanation uses mathematical language and interprets the equation in context.

   

  Student Project: Investigating a Real-World Function

  Project Goal

  Investigate a real-world relationship, collect or obtain data, construct a mathematical model, and evaluate how well the model describes the situation.

  Project Instructions

  Choose a research question

  Select a situation involving two measurable variables.

  Examples:

  • How does distance travelled change with time?

  • How does the height of a bouncing ball change between bounces?

  • How does the cost of a service change with usage?

  • How does the area of a square change with side length?

  • How does a hot drink's temperature change over time?

  • How does the length of a shadow change during the day?

  Identify the variables

  Clearly state:

  • Independent variable.

  • Dependent variable.

  • Units.

  • Relevant controlled conditions.

  Collect data

  Aim to collect at least 8–12 data points, where practical.

  Record the data in a clear table.

  If using a public dataset, identify the source.

  Create a scatter plot

  Label both axes with quantities and units.

  Use an appropriate scale.

  Choose a mathematical function

  Consider whether the relationship is linear, quadratic, exponential, inverse, periodic, or another type.

  Explain why the function was selected.

  Construct an equation

  Use calculations, graphing technology, or regression to determine a suitable equation.

  Explain the meaning of the coefficients.

  Analyse the model

  Identify relevant features, such as:

  • Intercepts.

  • Increasing and decreasing intervals.

  • Maximum and minimum values.

  • Domain and range.

  • End behaviour.

  • Rate of change.

  Not every feature will be relevant to every model.

  Evaluate accuracy

  Compare model predictions with observed data.

  Calculate residuals for selected points.

  Identify limitations.

  Make predictions

  Use the equation to make at least two predictions.

  Include one interpolation prediction and, where reasonable, one cautious extrapolation prediction.

  Communicate your findings

  Present your results as a written report, infographic, or presentation, depending on the assignment requirements.

  Suggested Project Report Structure

  Research Question

  State the relationship you are investigating.

  Example:

  "How does the distance travelled by a toy car change over time?"

  Hypothesis

  Predict the type of relationship.

  Example:

  "I predict that the distance travelled will increase approximately linearly with time if the toy car moves at a nearly constant speed."

  Variables and Method

  Identify the independent and dependent variables, equipment, and measurement procedure.

  Data Table

  Include all collected measurements and units.

  Graph

  Present a labelled scatter plot with the selected model.

  Mathematical Model

  State the equation and explain the coefficients.

  Analysis

  Discuss trends, intercepts, rates of change, extrema, and other relevant features.

  Predictions

  Show calculations and explain their meaning.

  Evaluation

  Discuss how well the model fits the data, its assumptions, and possible improvements.

  Conclusion

  Answer the research question using evidence from the model and observations.

  Example Project Assessment Rubric

  Suggested total: 20 marks

  Criterion	Marks	Expectations
  Data collection	4	Relevant, reliable data with appropriate units and method
  Model selection	4	Suitable function chosen and justified
  Equations and graphs	4	Correct equation, accurate graph, labelled axes
  Interpretation and evaluation	4	Meaningful analysis, predictions, limitations
  Mathematical communication	4	Clear explanations, correct terminology, logical presentation
  Total	20

  A high-quality project should demonstrate mathematical reasoning, not simply produce an attractive graph.

  Common Mistakes

  Choosing a function before examining the data

  The function should be selected based on the observed pattern and the context.

  Using too few data points

  A small dataset may not reveal the true relationship.

  Forgetting units

  A gradient may represent metres per second, dollars per hour, or another meaningful quantity.

  Confusing correlation with causation

  Two variables changing together does not necessarily mean that one causes the other to change.

  Assuming the model must pass through every point

  Real measurements often contain uncertainty.

  Using an equation without explaining its coefficients

  A mathematical model should have a meaningful interpretation.

  Making unrealistic extrapolations

  A relationship observed over a short interval may not continue indefinitely.

  Ignoring residuals

  Large or patterned residuals may indicate that the selected function is unsuitable.

  Confusing a prediction with a measurement

  Predicted values come from the model; measured values come from observations.

  Failing to evaluate limitations

  A strong report recognises assumptions and sources of uncertainty.

  Key Terms

  • Mathematical model: A mathematical representation of a real-world situation.

  • Modelling cycle: The process of developing, testing, and improving a mathematical model.

  • Independent variable: The input variable.

  • Dependent variable: The output variable.

  • Data point: A pair of measured or observed values.

  • Scatter plot: A graph displaying paired numerical observations.

  • Linear model: A model with a constant rate of change.

  • Quadratic model: A model based on a second-degree polynomial.

  • Exponential model: A model involving constant proportional growth or decay.

  • Regression: A statistical method for fitting a mathematical relationship to data.

  • Trendline: A line or curve representing the general pattern of data.

  • Residual: The difference between an observed value and a predicted value.

  • Interpolation: Estimating a value within the observed data range.

  • Extrapolation: Predicting a value outside the observed data range.

  • Domain: The set of permitted input values.

  • Range: The set of possible output values.

  • Gradient: The rate of change of a linear function.

  • Intercept: A point where a graph meets a coordinate axis.

  • Extremum: A maximum or minimum value.

  • Prediction: An estimated result obtained from a model.

  • Assumption: A condition accepted when constructing a model.

  • Limitation: A factor restricting the accuracy or usefulness of a model.

  • Validation: Checking whether a model describes observed reality adequately.

  • Mathematical communication: Explaining mathematical ideas using accurate terminology, equations, graphs, and reasoning.

  Key Takeaways

  • Mathematical modelling connects functions to real-world situations.

  • A useful model begins with a clear question and relevant data.

  • Independent and dependent variables must be identified.

  • Tables and scatter plots help reveal patterns.

  • Linear functions describe approximately constant rates of change.

  • Quadratic functions can describe parabolic relationships.

  • Exponential functions describe constant proportional growth or decay.

  • The appropriate model depends on the data and context.

  • Equations allow us to calculate values and make predictions.

  • Residuals help evaluate the accuracy of a model.

  • Interpolation estimates values within the observed range.

  • Extrapolation predicts beyond the observed range and requires caution.

  • Models should be evaluated for accuracy, assumptions, and limitations.

  • Mathematical communication is essential for explaining and justifying findings.

   

  Check Your Understanding

  Recall and Understanding

  1. Define mathematical modelling.

  2. Explain the difference between an independent variable and a dependent variable.

  3. Why is a scatter plot useful when selecting a mathematical model?

  4. Explain the difference between interpolation and extrapolation.

  5. What is a residual, and what does it tell us about a model?

  Application

  6. A taxi company charges a fixed fee of $4 plus $2.50 per kilometre. Write a linear model for the total cost and calculate the cost of a 12 km journey.

  7. A population begins at 300 and grows by 10% per year. Write an exponential model and calculate the population after 3 years.

  8. A ball's height is modeled by h(t) = −5t² + 20t + 2. Determine its maximum height and the time at which it occurs.

  9. A linear model predicts a value of 18.5, while the observed value is 17.2. Calculate the residual and interpret it.

  10. A student measures the side length and area of several squares. Which function family should they select, and why?

  Analysis and Challenge

  11. A dataset shows that a quantity increases by approximately 5 units every hour. Explain why a linear model may be appropriate.

  12. A dataset shows that a quantity doubles every 3 hours. Explain why an exponential model may be more appropriate than a linear model.

  13. A model describes a car travelling at a constant speed of 15 m/s. Explain why predicting its distance after 10 seconds is more reliable than predicting its distance after 10 hours if the original data covers only the first 20 seconds.

  14. A student chooses a quadratic model because it passes through every measured data point, even though the data follows an approximately straight-line trend. Explain why this may not be the best modelling decision.

  15. Design a brief mathematical modelling investigation involving a real-world relationship. Identify the research question, variables, expected function family, data collection method, and one limitation.

  Answers to Check Your Understanding

  1. Mathematical modelling is the process of representing real-world situations using mathematical relationships, equations, functions, tables, or graphs.

  2. The independent variable is the input, while the dependent variable is the output that changes in response to the input.

  3. A scatter plot reveals patterns in data and helps determine whether a linear, quadratic, exponential, or another model may be appropriate.

  4. Interpolation estimates values within the range of observed data. Extrapolation predicts values outside that range.

  5. A residual is the difference between an observed value and the model's prediction:

  Residual = Observed − Predicted

  Residuals help evaluate how accurately the model describes the data.

  6.

  Let d be distance in kilometres.

  Cost model:

  C(d) = 2.50d + 4

  For 12 km:

  C(12) = 2.50(12) + 4

  C(12) = 30 + 4

  Total cost = $34

  7.

  Initial population = 300

  Growth factor = 1.10

  Model:

  P(t) = 300(1.10)ᵗ

  After 3 years:

  P(3) = 300(1.10)³

  P(3) = 399.3

  Predicted population ≈ 399

  8.

  h(t) = −5t² + 20t + 2

  Vertex time:

  t = −20/[2(−5)]

  t = 2

  Calculate:

  h(2) = −5(2)² + 20(2) + 2

  h(2) = −20 + 40 + 2

  h(2) = 22

  Maximum height = 22 m at t = 2 s

  9.

  Residual = Observed − Predicted

  Residual = 17.2 − 18.5

  Residual = −1.3

  The model overestimated the observed value by 1.3 units.

  10.

  A quadratic model is appropriate because the area of a square is:

  A = s²

  where s is side length.

  The relationship between side length and area is quadratic.

  11.

  A linear model is appropriate because the quantity changes by approximately the same amount for each equal increase in time.

  This suggests a constant rate of change.

  12.

  An exponential model is appropriate because doubling over equal time intervals represents constant proportional growth.

  A linear model would increase by a constant amount rather than a constant factor.

  13.

  A prediction after 10 seconds lies within the observed interval and is interpolation.

  A prediction after 10 hours extends far beyond the observed interval and is extrapolation.

  The car may change speed, stop, or change direction over such a long period.

  Therefore, the 10-hour prediction is much less reliable.

  14.

  A model that passes through every measured point may be unnecessarily complicated or may fit measurement noise rather than the underlying relationship.

  If the data follows an approximately straight-line trend, a linear model may be simpler, easier to interpret, and more appropriate for predictions.

  15. Example investigation

  Research question:

  "How does the temperature of a hot drink change over time?"

  Independent variable:

  Time in minutes.

  Dependent variable:

  Temperature in °C.

  Expected function family:

  Exponential decay toward room temperature.

  Data collection:

  Measure the temperature every two minutes for 30 minutes using a thermometer.

  Possible limitation:

  The cooling rate may change because of air movement, the container, evaporation, or changes in surrounding temperature.

  A suitable model could take the form:

  T(t) = Tₐ + (T₀ − Tₐ)e^(−kt)

  where:

  • Tₐ = surrounding temperature.

  • T₀ = initial temperature.

  • k = positive cooling constant.

  • t = time.

  The parameters could be estimated from the measurements.

   

  Final Summary

  Mathematical modelling is a powerful method for connecting classroom mathematics with real-world problems.

  By collecting data, identifying patterns, selecting suitable functions, constructing equations, and evaluating predictions, we can use mathematics to understand relationships and make informed decisions.

  A successful modelling project requires more than finding an equation. It also requires explaining why the model is appropriate, interpreting its features, recognising limitations, and communicating conclusions clearly.

  The most important goal of mathematical modelling is not simply to produce an equation, but to use mathematics to explain and understand a real-world situation.