Quadratic Functions

A quadratic function is a function in which the highest power of the variable is 2.

The standard form of a quadratic function is:

y = ax² + bx + c

where:

• a, b, and c are constants
• a ≠ 0
• x is the independent variable
• y is the dependent variable

Examples of quadratic functions include:

y = x²

y = 2x² + 3x − 4

y = −x² + 6x + 2

y = 5x² − 20

The graph of every quadratic function is a curved shape called a parabola.

 

 

 

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Recognizing a Quadratic Function

One of the easiest ways to recognize a quadratic function is to look at its equation.

Consider:

y = 3x² + 2x − 5

The highest power of x is:

2

Therefore the function is quadratic.

Compare:

y = 3x + 2

The highest power is:

1

so this function is linear, not quadratic.

The General Form

A quadratic function can be written:

y = ax² + bx + c

The value of a cannot equal zero.

Why?

If:

a = 0

then:

y = 0x² + bx + c

which simplifies to:

y = bx + c

That is a linear function.

Therefore:

a ≠ 0

is necessary for a function to be quadratic.

Examples and Non-Examples

Quadratic

y = x² + 4x + 3

y = −2x² + 5

y = 7x²

y = 0.5x² − 3x + 8

Not Quadratic

y = 3x + 7 — linear

y = x³ − 2x — cubic

y = 1/x — reciprocal relationship

y = 2ˣ — exponential

The defining feature of a quadratic polynomial is that its highest variable exponent is 2.

What Is a Parabola?

The graph of a quadratic function is called a parabola.

A parabola has a characteristic curved shape.

It may:

• open upward
• open downward
• be narrow
• be wide
• move left or right
• move up or down

But it always has the basic parabolic shape.

Unlike a straight line, its steepness changes continuously.

Upward-Opening Parabolas

Consider:

y = x²

Its graph opens upward.

The lowest point occurs at:

(0, 0)

This point is called the vertex.

For quadratic functions:

a > 0 → parabola opens upward

An upward-opening parabola has a minimum value.

 

 

 

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Downward-Opening Parabolas

Consider:

y = −x²

Its graph opens downward.

For quadratic functions:

a < 0 → parabola opens downward

A downward-opening parabola has a maximum value.

This shape commonly appears when modeling objects moving through the air.

The Leading Coefficient

In:

y = ax² + bx + c

the coefficient a is called the leading coefficient.

Its sign tells us the direction in which the parabola opens.

If:

a > 0

the parabola opens upward.

If:

a < 0

the parabola opens downward.

The size of |a| also affects the shape.

Width of a Parabola

Compare:

y = x²

and:

y = 4x²

The graph of:

y = 4x²

is narrower.

Compare:

y = x²

and:

y = 0.25x²

The graph of:

y = 0.25x²

is wider.

In general:

• larger |a| → narrower parabola
• smaller |a| between 0 and 1 → wider parabola

The sign of a determines direction.

The magnitude of a affects width.

 

 

 

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The Vertex

Every parabola has a special point called the vertex.

The vertex is the turning point of the graph.

For an upward-opening parabola, the vertex is the:

minimum point

For a downward-opening parabola, the vertex is the:

maximum point

For:

y = x²

the vertex is:

(0, 0)

Maximum and Minimum Values

Consider:

y = (x − 3)² + 2

The parabola opens upward.

Its vertex is:

(3, 2)

Therefore its minimum y-value is:

2

Now consider:

y = −(x − 3)² + 8

The parabola opens downward.

Its vertex is:

(3, 8)

Therefore its maximum y-value is:

8

Axis of Symmetry

Parabolas are symmetrical.

A vertical line passing through the vertex divides the parabola into two mirror-image halves.

This line is called the axis of symmetry.

For:

y = x²

the axis of symmetry is:

x = 0

For a parabola with vertex:

(3, 2)

the axis of symmetry is:

x = 3

 

 

 

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Symmetrical Points

Consider:

y = x²

If:

x = 2

then:

y = 4

If:

x = −2

then:

y = 4

Therefore:

(2, 4)

and:

(−2, 4)

are symmetrical points.

They are the same distance from the axis of symmetry.

Quadratic Functions in Tables

Quadratic relationships can also be recognized from tables.

Consider:

These values follow:

y = x²

Notice that the y-values do not change by a constant amount.

Therefore this is not linear.

First Differences

Consider:

Calculate the first differences:

+1, +3, +5, +7, +9

They are not constant.

This tells us the function is not linear.

Second Differences

Now calculate the differences between the first differences:

First differences:

1, 3, 5, 7, 9

Second differences:

2, 2, 2, 2

The second differences are constant.

For equally spaced x-values:

constant second differences are a key sign of a quadratic relationship.

Another Quadratic Table

Consider:

First differences:

3, 7, 11, 15

Second differences:

4, 4, 4

The constant second difference indicates a quadratic relationship.

Linear Versus Quadratic Functions

A linear function has the general form:

y = mx + b

A quadratic function has the general form:

y = ax² + bx + c

The most important difference is their rate of change.

A linear function has a:

constant rate of change

A quadratic function has a:

changing rate of change

Comparing Their Graphs

A linear function produces:

a straight line

A quadratic function produces:

a parabola

 

 

 

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A straight line does not bend because its slope remains constant.

A parabola curves because its slope changes as x changes.

Comparing Tables

Linear Example

First differences:

+3, +3, +3, +3

Constant first differences indicate a linear relationship.

Quadratic Example

First differences:

+1, +3, +5, +7

Second differences:

+2, +2, +2

Constant second differences indicate a quadratic relationship.

A Useful Comparison

Rate of Change in a Quadratic

Consider:

y = x²

From x = 0 to x = 1:

y increases by:

1

From x = 1 to x = 2:

y increases by:

3

From x = 2 to x = 3:

y increases by:

5

From x = 3 to x = 4:

y increases by:

7

The rate of change keeps increasing.

This changing rate creates the curved graph.

Quadratics Can Increase and Decrease

A linear function with positive slope increases continuously.

A parabola can behave differently.

For:

y = x²

as x moves toward 0 from negative values, y decreases.

After x = 0, y increases.

So the function:

decreases → reaches a minimum → increases

The turning point is the vertex.

Downward Parabolas

For:

y = −x²

the behaviour is reversed.

The function:

increases → reaches a maximum → decreases

This is why downward-opening parabolas are useful for situations involving quantities that rise to a maximum and then fall.

Quadratic Relationships in the Real World

Quadratic relationships appear in many situations, including:

• projectile motion
• falling objects
• area problems
• optimization
• revenue and profit
• engineering
• architecture
• satellite dishes
• headlights and reflectors

 

 

 

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Projectile Motion

Imagine a ball thrown upward and forward.

Its height might be modeled by:

h = −5t² + 20t + 2

where:

• h = height
• t = time

This is quadratic because it contains:

t²

The negative coefficient of t² tells us the graph opens downward.

The ball:

• begins at an initial height
• rises
• reaches a maximum height
• falls

The graph forms a parabola.

Interpreting a Projectile Graph

On a height-time graph:

y-intercept

can represent the initial height.

vertex

can represent the maximum height.

x-intercept

can represent the time when the object reaches ground level.

Different features of the parabola have real physical meanings.

Area as a Quadratic Relationship

Suppose a rectangle has dimensions:

x

and:

x + 4

Its area is:

A = x(x + 4)

Expand:

A = x² + 4x

This is quadratic.

The area does not increase at a constant rate as x increases because both dimensions affect the area.

Square Area

For a square:

A = s²

If the side length doubles:

s → 2s

then:

A → (2s)²

A → 4s²

The area becomes four times as large.

This is a quadratic relationship, not a linear one.

Comparing Linear and Quadratic Growth

Consider:

Linear: y = 4x

Quadratic: y = x²

For small x-values, the linear function may be larger.

For sufficiently large positive x-values, x² eventually grows faster.

For example:

The functions are equal at x = 4 in this positive range, and after that the quadratic grows faster.

Quadratic Revenue Models

Suppose a business finds that revenue depends on price according to:

R = −20p² + 600p

This quadratic opens downward.

Therefore revenue increases up to a certain point and then decreases.

The vertex represents the price associated with the maximum revenue in the model.

This makes quadratic functions useful in optimization problems.

Why Revenue Might Be Quadratic

Increasing a product's price can increase the money earned per sale.

But a higher price may also reduce the number of products sold.

When these effects interact, revenue may form a quadratic relationship.

This illustrates why real situations do not always have constant rates of change.

Parabolic Reflectors

Satellite dishes and some reflectors use parabolic shapes.

A parabola has useful reflective properties.

Incoming signals travelling parallel to the axis can be directed toward a point called the focus.

This makes parabolic shapes useful in:

• satellite dishes
• radio telescopes
• some headlights
• reflecting telescopes

The physical shape of the parabola has practical engineering applications.

Recognizing a Quadratic from an Equation

Ask:

What is the highest power of the variable?

If the expression is a polynomial and the highest power is:

2

then it is quadratic.

Examples:

f(x) = 7x² − 4x + 1 → quadratic

f(x) = x³ + x² → not quadratic

f(x) = 5x − 8 → not quadratic

Recognizing a Quadratic from a Graph

Ask:

Is the graph parabolic?

Look for:

• a smooth curved shape
• one turning point
• symmetry around a vertical line
• upward or downward opening

These features strongly indicate a quadratic function.

Recognizing a Quadratic from a Table

If x-values are equally spaced:

1. calculate first differences
2. calculate second differences

If:

first differences are not constant

but:

second differences are constant

the relationship is quadratic.

Recognizing a Quadratic from Context

Look for situations involving:

• squared quantities
• areas
• maximum or minimum values
• projectile height
• products of changing quantities
• increasing or decreasing rates

These may suggest a quadratic relationship.

However, context alone does not prove that a relationship is quadratic. The equation or data should support the conclusion.

Different Forms of Quadratic Functions

Quadratic functions can be written in several forms.

Standard Form

y = ax² + bx + c

Useful for identifying:

• leading coefficient
• y-intercept

Vertex Form

y = a(x − h)² + k

Useful for identifying:

• vertex (h, k)
• maximum or minimum
• axis of symmetry

Factored Form

y = a(x − r₁)(x − r₂)

Useful for identifying:

• x-intercepts or roots

These forms represent the same type of function.

The y-Intercept

In standard form:

y = ax² + bx + c

set:

x = 0

Then:

y = c

Therefore the y-intercept is:

(0, c)

For:

y = 2x² − 3x + 5

the y-intercept is:

(0, 5)

The x-Intercepts

The x-intercepts occur where:

y = 0

For example:

y = x² − 4

Set y = 0:

0 = x² − 4

x² = 4

x = 2 or x = −2

Therefore the x-intercepts are:

(−2, 0) and (2, 0)

A quadratic may have:

• two x-intercepts
• one x-intercept
• no real x-intercepts

Parabolas and Symmetry

Consider:

y = (x − 3)² + 1

Its vertex is:

(3, 1)

Its axis of symmetry is:

x = 3

Points the same horizontal distance from x = 3 have the same y-value.

For example, x = 2 and x = 4 are both one unit from the axis.

Therefore their y-values are equal.

This symmetry is a defining feature of parabolas.

Comparing Two Quadratics

Consider:

A: y = x²

B: y = −2x²

Both are quadratic.

A:

• opens upward
• has a minimum

B:

• opens downward
• has a maximum
• is narrower because |−2| > |1|

The coefficient a gives important information about the shape.

Is Every Curved Graph Quadratic?

No.

Many other functions produce curved graphs.

Examples include:

• exponential functions
• cubic functions
• reciprocal functions
• circles
• logarithmic functions

A quadratic graph specifically has a parabolic shape.

Do not classify every curved graph as quadratic.

Is Every Quadratic Graph the Same Shape?

All quadratic graphs are parabolas, but they can differ in:

• width
• direction
• vertex location
• intercepts
• maximum or minimum value

For example:

y = x²

y = 5x²

y = −x²

y = (x − 4)² + 2

are all quadratic functions, but their graphs are positioned or shaped differently.

A Complete Recognition Strategy

When deciding whether a relationship is quadratic, use:

Equation → Graph → Table → Context

Equation

Look for a polynomial with highest power 2.

Graph

Look for a parabola.

Table

For equally spaced x-values, look for constant second differences.

Context

Look for changing rates, squared quantities, maxima, minima, or other situations that can produce quadratic relationships.

The more forms of evidence that agree, the more confident you can be.

Common Mistakes

Mistake: Thinking any equation containing x² must automatically be quadratic.

The highest relevant power must be 2 after simplification.

For example:

x² + x³

is cubic, not quadratic.

Mistake: Thinking all curved graphs are quadratic.

Quadratic graphs specifically have a parabolic shape.

Mistake: Looking for constant first differences in quadratic tables.

Linear functions have constant first differences.

Quadratic functions have constant second differences when x-values are equally spaced.

Mistake: Assuming all parabolas open upward.

If a < 0, the parabola opens downward.

Mistake: Assuming a negative a means the entire function is negative.

It means the parabola opens downward. Parts of the graph can still have positive y-values.

Mistake: Confusing the vertex with an intercept.

The vertex is the turning point. It does not necessarily lie on either axis.

Mistake: Thinking a quadratic has a constant slope.

Its rate of change varies.

Mistake: Assuming a quadratic always has two x-intercepts.

It may have zero, one, or two real x-intercepts.

Mistake: Thinking the y-intercept must be zero.

In:

y = ax² + bx + c

the y-intercept is:

(0, c)

Mistake: Ignoring context.

A quadratic model may be mathematically correct but only meaningful over a particular range.

Key Terms

Quadratic function — polynomial function whose highest variable power is 2.

Standard form — quadratic written as y = ax² + bx + c, where a ≠ 0.

Parabola — curved graph produced by a quadratic function.

Leading coefficient — coefficient a multiplying x².

Vertex — turning point of a parabola.

Maximum — greatest value of a downward-opening quadratic.

Minimum — least value of an upward-opening quadratic.

Axis of symmetry — vertical line dividing a parabola into mirror-image halves.

y-intercept — point where the graph crosses the y-axis.

x-intercept — point where the graph crosses the x-axis.

Root — x-value for which the quadratic equals zero.

Zero — another name for an x-value where the function equals zero.

First difference — change between consecutive y-values.

Second difference — change between consecutive first differences.

Rate of change — amount one variable changes relative to another.

Linear function — function with constant rate of change and a straight-line graph.

Quadratic relationship — relationship represented by a quadratic function.

Projectile — object moving through the air whose height can often be approximated by a quadratic function.

Quadratic model — quadratic function used to represent a real situation.

Optimization — finding a maximum or minimum value.

Key Takeaways

• A quadratic function has highest power 2.
• Its standard form is y = ax² + bx + c, where a ≠ 0.
• The graph of a quadratic function is a parabola.
• If a > 0, the parabola opens upward.
• If a < 0, the parabola opens downward.
• The magnitude of a affects the width of the parabola.
• Every parabola has a vertex.
• An upward-opening parabola has a minimum.
• A downward-opening parabola has a maximum.
• Every parabola has an axis of symmetry passing through its vertex.
• Points on opposite sides of the axis can have equal y-values.
• Linear functions have constant rates of change.
• Quadratic functions have changing rates of change.
• Linear graphs are straight lines.
• Quadratic graphs are parabolas.
• Linear tables have constant first differences when x-values are equally spaced.
• Quadratic tables have constant second differences when x-values are equally spaced.
• The y-intercept of y = ax² + bx + c is (0, c).
• Quadratics may have zero, one, or two real x-intercepts.
• Quadratic relationships occur in projectile motion, area, optimization, business, science, and engineering.
• A quadratic model can represent a quantity that increases to a maximum and then decreases.
• It can also represent a quantity that decreases to a minimum and then increases.
• Not every curved graph is quadratic.
• Equations, graphs, tables, and context can all provide evidence that a relationship is quadratic.
• Understanding the shape of a parabola helps us interpret real-world quadratic relationships.

Check Your Understanding

1. What is the general standard form of a quadratic function?
2. Why must a ≠ 0 in y = ax² + bx + c?
3. Which of these is quadratic: y = 4x + 3, y = 4x² + 3, or y = 4x³ + 3?
4. What is the name of the graph of a quadratic function?
5. Describe the graph of y = x².
6. Describe the graph of y = −x².
7. What does the sign of a tell you about a parabola?
8. Compare the widths of y = x² and y = 5x².
9. What is the vertex of y = x²?
10. Explain the difference between a maximum and minimum on a quadratic graph.
11. Explain what the axis of symmetry represents.
12. Determine whether the table (0, 2), (1, 5), (2, 8), (3, 11) is linear or quadratic. Explain.
13. Determine whether the table (0, 0), (1, 1), (2, 4), (3, 9), (4, 16) is linear or quadratic using differences.
14. Explain the main difference between the rates of change of linear and quadratic functions.
15. Compare the graphs of a linear function and a quadratic function.
16. A ball's height is modeled by h = −5t² + 20t + 2. Explain how you know the relationship is quadratic and why the parabola opens downward.
17. For a projectile height graph, explain what the vertex, y-intercept, and a positive x-intercept could represent.
18. A rectangle has dimensions x and x + 6. Write an expression for its area and explain why the area is a quadratic function of x.
19. A table has equally spaced x-values, nonconstant first differences, and constant second differences. What type of relationship does this suggest? Explain.
20. Give a real-world example of a quadratic relationship. Identify what the vertex could represent and explain why a linear model would not describe the situation as well.

This is why examining rates of change and data patterns is important instead of relying only on the general appearance of a graph.

Choosing the Best Model

Real-world data rarely announces which mathematical function should be used.

We need to examine how the quantity changes.

Ask:

Does it change by approximately the same amount?

A linear model may be appropriate.

Does the rate of change itself change steadily, producing approximately constant second differences?

A quadratic model may be appropriate.

Does it change by approximately the same percentage or multiplying factor?

An exponential model may be appropriate.

Real-World Example: Taxi Fare

Suppose a taxi charges:

$4 starting fee

plus

$2 per kilometre.

The cost increases by the same amount for every kilometre travelled.

A suitable model is:

C = 2d + 4

This is a linear function.

Real-World Example: Projectile Motion

Imagine throwing a ball upward.

Its height may increase, reach a maximum, and then decrease.

 

 

 

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Ignoring air resistance and using an appropriate coordinate system, the height can often be modeled using a quadratic function such as:

h = −4.9t² + vt + h₀

The graph is a downward-opening parabola.

Therefore, a quadratic model is appropriate.

Real-World Example: Population Growth

Suppose a bacterial population doubles every hour.

100 → 200 → 400 → 800 → 1600

The population does not increase by a constant amount.

Instead, it increases by a constant factor.

A possible model is:

P = 100(2ᵗ)

This is an exponential model.

 

 

 

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Real-World Example: Depreciation

Suppose a machine loses 15% of its value each year.

Its value is repeatedly multiplied by:

1 − 0.15 = 0.85

If its initial value is $20,000:

V = 20000(0.85)ᵗ

Because the value changes by a constant percentage rather than a constant amount, the relationship is exponential decay.

Comparing the Three Function Types

How to Identify a Function

When given a graph, table, equation, or real-world situation, use several clues rather than relying on only one.

Look at the equation

x → likely linear

x² → likely quadratic

variable in the exponent → likely exponential

Examples:

y = 4x + 7 → linear

y = 3x² − 2x + 1 → quadratic

y = 5(1.2)ˣ → exponential

Look at the graph

Straight line → linear

Parabola → quadratic

Growth or decay curve → possibly exponential

Look at a table

Constant first difference → linear

Constant second difference → quadratic

Constant ratio → exponential

Look at the situation

Constant amount of change → linear

Parabolic or steadily changing rate → quadratic

Constant percentage or multiplying factor → exponential

Using several clues makes your classification more reliable.

Common Mistakes

Thinking every curved graph is quadratic

Exponential graphs are also curved. Look at the shape, equation, and rate of change.

Thinking exponential means simply "very fast"

Exponential describes a particular mathematical pattern involving repeated multiplication. Some exponential functions initially change quite slowly.

Confusing constant difference with constant ratio

Adding 5 repeatedly suggests linear growth.

Multiplying by 5 repeatedly suggests exponential growth.

Checking second differences before checking first differences

A linear sequence also has constant second differences—specifically, zero. Check first differences first. If they are constant, classify the relationship as linear.

Using differences when x-values are not equally spaced

The simple first- and second-difference tests assume equally spaced x-values.

Assuming a real-world model is perfect

Mathematical functions are models. Real populations, prices, profits, and physical systems may only approximately follow a particular function over a limited range.

Key Terms

Linear Function: A function with a constant rate of change and a straight-line graph.

Quadratic Function: A function whose highest variable power is 2 and whose graph is a parabola.

Exponential Function: A function in which the variable appears in the exponent and equal input changes produce a constant multiplying factor.

Rate of Change: The amount an output changes compared with a change in input.

First Difference: The difference between consecutive output values in a table with equally spaced inputs.

Second Difference: The difference between consecutive first differences.

Constant Ratio: A pattern in which consecutive outputs are multiplied by the same factor.

Slope: The constant rate of change of a linear function.

Parabola: The characteristic U-shaped or inverted U-shaped graph of a quadratic function.

Vertex: The turning point of a parabola.

Exponential Growth: A pattern in which a quantity increases repeatedly by the same factor or percentage over equal intervals.

Exponential Decay: A pattern in which a quantity decreases repeatedly by the same factor or percentage over equal intervals.

Mathematical Model: A mathematical representation used to describe or predict a real-world relationship.

Check Your Understanding

1. Which type of function has a constant rate of change?

2. Which type of function produces a parabola?

3. Which type of function usually has a constant ratio between consecutive outputs when inputs increase equally?

4. Identify the function type:

4, 7, 10, 13, 16

5. Identify the function type:

1, 4, 9, 16, 25

6. Identify the function type:

3, 9, 27, 81, 243

7. A worker earns $18 for every hour worked. Which function type would best model total earnings versus hours?

8. A ball is thrown upward and then falls back toward the ground. Which function type could model its height versus time?

9. An investment grows by 6% each year. Which function type would best model its value?

10. Explain the difference between a constant difference and a constant ratio.

11. Why can looking only at the shape of a graph sometimes make it difficult to distinguish quadratic and exponential functions?

12. Describe one real-world example each of a linear, quadratic, and exponential relationship.

Key Takeaways

• Linear, quadratic, and exponential functions have different equations, graphs, and rates of change.
• Linear functions have a constant first difference when inputs are equally spaced.
• Linear functions produce straight-line graphs.
• Quadratic functions have constant second differences when inputs are equally spaced.
• Quadratic functions produce parabolic graphs.
• Exponential functions have a constant ratio for equal input intervals.
• Exponential relationships involve repeated multiplication rather than repeated addition.
• Constant amounts of change often suggest a linear model.
• Constant percentages or multiplying factors often suggest an exponential model.
• Situations involving parabolic motion or steadily changing rates may be modeled with quadratic functions.
• Tables, graphs, equations, and the context of a situation should all be considered when selecting a model.
• Choosing the best function type helps us describe patterns, make predictions, and model real-world relationships.

Modeling with Functions

A mathematical model uses mathematics to represent a real-world situation. Functions are especially useful because they describe how one quantity changes in relation to another.

For example, functions can model:

• The cost of a taxi journey
• The height of a ball after it is thrown
• The growth of a population
• The value of an investment
• The temperature of an object over time
• The amount of fuel used during a journey

The goal is not simply to find an equation. We need to choose a function that reasonably represents the pattern in the real situation.

 

 

 

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What Is a Function Model?

A function model is a function used to represent the relationship between two quantities.

One quantity is usually the independent variable.

The other is the dependent variable because its value depends on the independent variable.

Suppose a bicycle rental company charges $8 plus $4 per hour.

Let:

t = rental time in hours

C = total cost in dollars

A model is:

C = 4t + 8

The model allows us to calculate the cost for different rental times.

For 3 hours:

C = 4(3) + 8

C = 20

The predicted cost is $20.

Choosing an Appropriate Function Type

Different situations require different types of functions.

Three common models are:

• Linear
• Quadratic
• Exponential

The pattern in the data and the context of the situation help determine which model is appropriate.

Linear Models

A linear model is appropriate when a quantity changes at approximately a constant rate.

A common form is:

y = mx + b

where:

m = constant rate of change

b = initial value

Consider a water tank containing 100 L of water. Water enters at a constant rate of 20 L per minute.

A model is:

V = 20t + 100

where:

V = volume in litres

t = time in minutes

The model is linear because the volume increases by the same amount every minute.

 

 

 

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Clues That Suggest a Linear Model

Look for:

• Constant rate of change
• Constant first differences
• Approximately straight-line data
• The same amount being added or removed repeatedly

Examples include constant speed, hourly wages, constant filling rates and fixed cost per item.

Quadratic Models

A quadratic model is useful when the rate of change itself changes in a regular way.

A common form is:

y = ax² + bx + c

Its graph is a parabola.

One common example is projectile motion.

Imagine a ball thrown upward from a height of 2 m.

Its height might be modeled approximately by:

h = −4.9t² + 15t + 2

where:

h = height in metres

t = time in seconds

The ball initially rises, reaches a maximum height and then falls.

 

 

 

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Clues That Suggest a Quadratic Model

Look for:

• A parabolic graph
• A quantity that increases and then decreases
• A quantity that decreases and then increases
• Constant second differences for equally spaced inputs
• Situations involving certain types of area or projectile motion

Exponential Models

An exponential model is appropriate when a quantity changes by approximately the same factor or percentage over equal intervals.

A common form is:

y = abˣ

where:

a = initial value

b = growth or decay factor

Suppose a bacterial population begins with 500 bacteria and doubles every hour.

A model is:

P = 500(2ᵗ)

After 1 hour:

P = 500(2¹) = 1000

After 2 hours:

P = 500(2²) = 2000

After 3 hours:

P = 500(2³) = 4000

The population increases by different amounts each hour, but it always multiplies by the same factor.

 

 

 

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Clues That Suggest an Exponential Model

Look for:

• Constant percentage change
• Constant multiplying factor
• Constant ratio for equally spaced inputs
• Rapidly increasing growth
• Exponential decay toward zero

Examples include simplified population growth, compound interest, radioactive decay and depreciation.

Comparing the Models

The choice of model depends on how the quantity changes.

A useful question is:

What stays constant?

If the difference stays constant → think linear.

If the second difference stays constant → think quadratic.

If the ratio or percentage change stays constant → think exponential.

Choosing a Model from Data

Suppose a company records the following sales:

The differences are:

+20, +20, +20, +20

Therefore, a linear model is appropriate.

Now consider:

Each value is approximately:

previous value × 1.2

This represents growth of 20% per month.

Therefore, an exponential model is appropriate.

Justifying the Choice of Model

Choosing a model is only part of the process. We should also be able to explain why the model is appropriate.

A strong justification identifies the mathematical pattern.

For example:

A linear model is appropriate because the distance increases by approximately the same amount during each equal time interval.

Or:

An exponential model is appropriate because the population increases by approximately the same percentage each year.

A weak justification would simply say:

It looks exponential.

Instead, identify the evidence that supports the choice.

Using Models to Make Predictions

Once a model has been created, it can be used to estimate unknown values.

Suppose the number of visitors to a website is modeled by:

V = 500 + 75t

where t represents the number of months.

Predict the number of visitors after 8 months.

Substitute:

t = 8

V = 500 + 75(8)

V = 500 + 600

V = 1100

The model predicts approximately 1100 visitors after 8 months.

The word approximately is important because a mathematical model may not perfectly predict what actually happens.

Interpreting Predictions

A calculated answer should always be interpreted in the context of the problem.

Suppose a population model gives:

P(10) = 24,580

Do not simply report:

24,580

Instead explain:

The model predicts that the population will be approximately 24,580 after 10 years.

A complete interpretation should identify:

• What the number represents
• The appropriate units
• The input value used
• Whether the answer is exact or a prediction

Interpolation and Extrapolation

Predictions can be made within or outside the range of observed data.

Interpolation

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

Suppose measurements were collected between 0 and 10 minutes.

Using the model to estimate a value at 6 minutes is interpolation.

Interpolation is often relatively reliable because the prediction lies within the region where data were collected.

Extrapolation

Extrapolation means predicting beyond the range of observed data.

Using the same data to predict what happens after 100 minutes would be extrapolation.

 

 

 

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Extrapolation is generally more uncertain because the relationship may change outside the observed range.

Models Have Limitations

Mathematical models simplify reality.

A model may work very well under certain conditions but poorly under others.

For example, suppose a town has a population of 10,000 and grows by 5% every year.

An exponential model might be:

P = 10000(1.05)ᵗ

The model assumes that the population continues growing at exactly 5% indefinitely.

In reality, population growth can be affected by:

• Housing availability
• Jobs
• Migration
• Birth rates
• Death rates
• Government policies
• Economic conditions
• Natural disasters

Therefore, the model may be useful for short-term predictions but unreliable far into the future.

Models May Produce Impossible Predictions

One way to evaluate a model is to ask:

Does the prediction make sense in the real world?

Suppose a water tank contains 100 L and drains at 5 L per minute.

A linear model might be:

V = 100 − 5t

After 10 minutes:

V = 50 L

This makes sense.

After 20 minutes:

V = 0 L

This also makes sense.

But after 30 minutes:

V = −50 L

A tank cannot contain −50 L of water.

The mathematics continues, but the real-world model should stop when the tank reaches zero.

Therefore, the useful domain of the model is:

0 ≤ t ≤ 20

This illustrates why the domain of a mathematical model is often restricted by the real situation.

Assumptions in Models

Models are usually based on assumptions.

An assumption is something we accept as true in order to simplify a situation.

For example, a constant-speed travel model may assume:

• The vehicle maintains the same speed
• There are no stops
• Traffic conditions do not change
• The route does not change

These assumptions make the mathematics easier.

However, they also create limitations.

If the assumptions become unrealistic, the model becomes less reliable.

Worked Example: Taxi Fare

A taxi company charges a $5 starting fee plus $3 for every kilometre travelled.

Choose a Model

The fare increases by a constant amount for every kilometre.

Therefore, a linear model is appropriate.

Let:

d = distance travelled

C = total cost

The model is:

C = 3d + 5

Make a Prediction

Find the cost of travelling 12 km.

C = 3(12) + 5

C = 36 + 5

C = $41

Interpret the Result

The model predicts that a 12 km taxi journey will cost $41.

Evaluate the Model

The model assumes that:

• The rate remains $3/km
• There are no additional fees
• There are no waiting-time charges
• The pricing system does not change

If these assumptions are incorrect, the actual fare may differ.

Worked Example: Depreciation

A computer is purchased for $2,000 and loses 20% of its value each year.

Because the value changes by the same percentage, an exponential model is appropriate.

The yearly multiplier is:

1 − 0.20 = 0.80

Therefore:

V = 2000(0.80)ᵗ

After 3 years:

V = 2000(0.80)³

V = 2000(0.512)

V = 1024

The model predicts a value of approximately $1,024 after three years.

 

 

 

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Limitation

The object will probably not continue losing exactly 20% of its value forever.

Eventually, other factors such as condition, rarity, demand or minimum resale value may become more important.

Worked Example: Projectile Motion

Suppose the height of a ball is modeled by:

h = −5t² + 20t + 1

where:

h = height in metres

t = time in seconds

Find its height after 2 seconds.

h = −5(2²) + 20(2) + 1

h = −20 + 40 + 1

h = 21 m

The model predicts that the ball is approximately 21 metres above the reference level after 2 seconds.

This model is quadratic because the variable t is squared.

Limitation

The model may assume simplified gravitational motion and may ignore effects such as air resistance and wind.

The Modeling Process

Function modeling can be viewed as a cycle.

 

 

 

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A useful process is:

Understand the situation → Identify variables → Examine the pattern → Choose a function → Build the model → Make predictions → Interpret results → Evaluate the model

If the model does not fit the data or produces unrealistic predictions, it may need to be changed.

Evaluating Whether a Model Is Good

A useful model should:

• Fit the available data reasonably well
• Produce realistic predictions
• Match the expected mathematical pattern
• Have reasonable assumptions
• Be useful within an appropriate domain

Ask questions such as:

Does the model fit the observed data?

Does the prediction make sense?

Are the assumptions reasonable?

How far beyond the data are we predicting?

Could another function provide a better model?

A model does not need to be perfect to be useful.

Common Mistakes

Choosing a function only because the graph looks similar

Also examine differences, ratios, rates of change and the context.

Assuming models make exact predictions

Most real-world models provide estimates rather than guaranteed outcomes.

Ignoring units

A prediction such as "35" is incomplete if we do not know whether it represents dollars, kilometres, people or seconds.

Extrapolating too far

A model that fits current data may become unreliable far beyond the observed range.

Ignoring unrealistic results

A mathematically correct calculation can still produce a physically impossible prediction.

Forgetting assumptions

Every model simplifies reality. Understanding its assumptions helps us judge when the model is useful.

Key Terms

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

Function Model: A function used to represent the relationship between quantities.

Independent Variable: The input variable used to determine another quantity.

Dependent Variable: The output variable whose value depends on the input.

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

Quadratic Model: A model represented by a quadratic function, often producing a parabolic graph.

Exponential Model: A model involving repeated proportional or percentage change.

Prediction: A value estimated using a mathematical model.

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

Extrapolation: Predicting a value beyond the range of observed data.

Assumption: A condition accepted as true when constructing a model.

Limitation: A condition or weakness that restricts how accurately or widely a model can be used.

Domain: The set of input values for which a model is meaningful.

Rate of Change: The amount an output changes relative to a change in input.

Check Your Understanding

1. What is a mathematical model?

2. A worker earns $20 per hour. Which function type would be appropriate for modeling earnings versus time? Explain your choice.

3. A population increases by 4% each year. Which type of function would be appropriate? Why?

4. A ball rises, reaches a maximum height and falls. Which type of function could model its height versus time?

5. A tank contains 300 L and loses water at 10 L per minute. Write a function that models the amount of water remaining.

6. Use your model from Question 5 to predict the amount remaining after 12 minutes.

7. Why would the model in Question 5 eventually stop being useful?

8. Explain the difference between interpolation and extrapolation.

9. Why are predictions made far beyond the available data generally less reliable?

10. A laptop worth $1,500 loses 10% of its value each year. Write an exponential model for its value after t years.

11. Use the model to predict the laptop's value after 2 years.

12. Give one limitation of using the same depreciation model for the next 30 years.

Key Takeaways

• A mathematical model uses mathematics to represent a real-world relationship.
• Linear models are appropriate when quantities change by approximately a constant amount.
• Quadratic models are useful for relationships involving parabolic behaviour or steadily changing rates.
• Exponential models are appropriate when quantities change by approximately a constant factor or percentage.
• A model should be chosen using mathematical evidence and the context of the situation.
• A good justification explains why the pattern matches the chosen function type.
• Models can be used to make predictions and solve practical problems.
• Predictions should always be interpreted using the correct context and units.
• Interpolation makes predictions within the observed data range.
• Extrapolation makes predictions beyond the observed data range and is usually less reliable.
• Every mathematical model involves assumptions and therefore has limitations.
• A mathematically correct prediction may still be unrealistic in the real world.
• The meaningful domain of a model may be smaller than the mathematical domain of its equation.
• Good mathematical modeling involves choosing, using, interpreting and then evaluating the model—not simply finding an equation.