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.

---

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.

---

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.

---

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

---

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

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

---

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.