Quadratic Graphs: Vertex, Intercepts and Range

Alberta Grade 11 Mathematics

Relations and Functions

Vertices, intercepts, domain, range, and symmetry

A quadratic graph has a turning point and matching sides. It may open upward like a bowl or downward like an upside-down bowl. The turning point, places where the graph meets the axes, and the values the graph can take are connected. This lesson shows how to describe those features from an equation and use them to understand a graph.

What you will learn

Identify a quadratic vertex and its axis of symmetry.
Describe the x- and y-intercepts of a quadratic graph.
State the domain and range of a quadratic function.
Use symmetry and the graph’s opening direction to check a description.

1. Prerequisite review: coordinates and intercepts

A point on a graph is written as an ordered pair, such as

(2,-3)

. The first number gives the horizontal position, called the

x

-coordinate. The second gives the vertical position, called the

y

-coordinate. The horizontal axis is the

x

-axis, and the vertical axis is the

y

-axis.

A quadratic function can be written as

y=ax^2+bx+c

, where

a

,

b

, and

c

are numbers and

a\ne 0

. The squared variable makes the equation quadratic. The sign of

a

tells which way the graph opens: it opens upward when

a>0

and downward when

a<0

.

An intercept is a point where a graph meets an axis. At an

x

-intercept, the output is zero, so

y=0

. At a

y

-intercept, the input is zero, so

x=0

. An intercept may be stated as a point, such as

(3,0)

, or by its coordinate when the meaning is clear.

y=ax^2+bx+c,\quad a\ne 0

Coordinates are written in the order $(x,y)$ .
Set $y=0$ to find x-intercepts.
Set $x=0$ to find the y-intercept.

2. Vertex, symmetry, domain, and range

The vertex is the turning point of a quadratic graph. It is the lowest point if the graph opens upward and the highest point if it opens downward. In vertex form,

y=a(x-h)^2+k

, the vertex is

(h,k)

. The sign inside the brackets can be easy to misread: for example,

(x-4)^2

gives an

x

-coordinate of

4

.

The axis of symmetry is a vertical line that divides the graph into matching left and right sides. The vertex lies on this line. If the vertex is

(h,k)

, the axis is

x=h

. Points the same horizontal distance from the axis have equal

y

-values.

For an equation in the form

y=ax^2+bx+c

, the axis can be found using

x=-\frac{b}{2a}

. This gives the vertex’s horizontal coordinate. Substitute that value for

x

in the equation to get the vertex’s vertical coordinate. The rule is useful when the equation is not already in vertex form.

The domain is the set of possible inputs, or

x

-values. A quadratic function has a value for every real input, so its domain is all real numbers. The range is the set of possible outputs, or

y

-values. The vertex and opening direction determine the range. An upward-opening graph has outputs at or above the vertex’s

y

-value. A downward-opening graph has outputs at or below it.

x=-\frac{b}{2a}

Vertex form gives the vertex directly.
The axis of symmetry passes through the vertex.
A quadratic function’s domain is all real numbers.
The vertex gives the minimum or maximum output and sets the boundary of the range.

3. Connect equations to a calculated graph

Consider

y=(x-2)^2-3

. This is vertex form, so the vertex is

(2,-3)

and the axis is

x=2

. The coefficient of the squared term is positive, so the graph opens upward. Its lowest output is

-3

, and its range is

y\ge -3

.

For the x-intercepts, set

y=0

. Then

(x-2)^2=3

, so

x=2-\sqrt{3}

or

x=2+\sqrt{3}

. For the y-intercept, set

x=0

; the output is

1

, giving the point

(0,1)

. The two x-intercepts are equally far from the axis

x=2

, as symmetry predicts.

The calculated figure uses this same equation. Its axes are labelled with

x

and

y

, and its curve is calculated from the coefficients. Use it to connect the vertex and intercept coordinates to the features of the graph. A sketch can help explain a result, but the equation gives the exact values.

y=(x-2)^2-3

Use the equation to calculate intercepts, then state them as points.
The x-intercepts in this example lie on opposite sides of the symmetry axis.
The vertex is included in the range.

4. Student try and reasonableness checks

Try describing

y=-(x-1)^2+4

. Identify the vertex and axis, find both intercepts, and state the domain and range. First use the negative coefficient to predict whether the vertex is a maximum or a minimum.

To check your description, compare the x-intercepts with the axis: when there are two intercepts, they should be the same distance from it. Check that the range includes the vertex’s output. Also make sure the range extends in the direction shown by the opening. For the quadratic function itself, the domain remains all real numbers.

A negative squared-term coefficient means the vertex is a maximum.
Check intercepts against the symmetry axis.
Check the range against both the vertex and opening direction.

Worked example

Read features from vertex form

Describe the vertex, axis of symmetry, intercepts, domain, and range of

y=(x-2)^2-3

.

Quadratic with minimum at (2, -3)

The graph opens upward, crosses the y-axis at (0, 1), and has two x-intercepts symmetric about x = 2.

Read the vertex
Compare the equation with vertex form. The bracket gives the horizontal coordinate, and the number outside gives the vertical coordinate.
$(h,k)=(2,-3)$
Find the intercepts
Set $y=0$ to find the x-intercepts. Set $x=0$ to find the y-intercept.
$x=2\pm\sqrt{3},\quad y(0)=1$
State domain and range
The positive coefficient means the graph opens upward. The vertex is its minimum, and every real number is an allowed input.
$\text{Domain: }\mathbb{R},\quad \text{Range: }y\ge -3$

Answer: The vertex is

(2,-3)

, and the axis of symmetry is

x=2

. The x-intercepts are

(2-\sqrt{3},0)

and

(2+\sqrt{3},0)

. The y-intercept is

(0,1)

. The domain is all real numbers, and the range is

y\ge -3

.

Check: The x-intercepts are equally far from

x=2

. The vertex’s output,

-3

, is included as the minimum of the range.

Worked example

Find features from expanded form

Describe the vertex, axis of symmetry, intercepts, domain, and range of

y=-2x^2-4x+6

.

Quadratic with maximum at (-1, 8)

The graph opens downward and has matching sides across x = -1.

Find the axis and vertex
Use the axis rule with $a=-2$ and $b=-4$ . Substitute the resulting $x$ -coordinate into the function to find the vertex’s output.
$x=-1,\quad y(-1)=8$
Find the intercepts
Set $x=0$ for the y-intercept. For the x-intercepts, set $y=0$ and factor the expression.
$y(0)=6,\quad -2(x+3)(x-1)=0$
Describe domain and range
The negative leading coefficient means the graph opens downward. The vertex is its maximum, while all real inputs are allowed.
$\text{Domain: }\mathbb{R},\quad \text{Range: }y\le 8$

Answer: The vertex is

(-1,8)

, and the axis of symmetry is

x=-1

. The x-intercepts are

(-3,0)

and

(1,0)

. The y-intercept is

(0,6)

. The domain is all real numbers, and the range is

y\le 8

.

Check: The x-intercepts are each two units from

x=-1

. Since the graph opens downward, the vertex output of

8

is the greatest output.

Worked example

Use intercepts to locate the axis

For

y=(x-3)(x+1)

, describe the vertex, axis of symmetry, intercepts, domain, and range.

Quadratic with minimum at (1, -4)

The x-intercepts at -1 and 3 are symmetric about x = 1.

Find the axis
The factors show that the x-intercepts are $3$ and $-1$ . By symmetry, the axis is halfway between these values.
$x=\frac{3+(-1)}{2}=1$
Find the vertex
Substitute the axis value into the function. The positive leading coefficient means this vertex is the minimum.
$y(1)=(1-3)(1+1)=-4$
Find the y-intercept and range
Set $x=0$ to find the y-intercept. The graph opens upward, so its outputs start at the vertex’s value and extend upward.
$y(0)=-3,\quad y\ge -4$

Answer: The vertex is

(1,-4)

, and the axis of symmetry is

x=1

. The x-intercepts are

(-1,0)

and

(3,0)

. The y-intercept is

(0,-3)

. The domain is all real numbers, and the range is

y\ge -4

.

Check: The roots are equally far from

x=1

. The minimum output,

-4

, is less than the y-intercept output,

-3

, which fits an upward-opening graph.

Common mistakes and how to avoid them

Reading the vertex of

y=(x-2)^2-3

as

(-2,-3)

.

Correction: The horizontal coordinate is the value that makes the bracket zero. The vertex is

(2,-3)

.

Calling the axis of symmetry a point.

Correction: The vertex is a point. The axis is a line, written

x=h

when the vertex’s horizontal coordinate is

h

.

Stating the range without checking which way the graph opens.

Correction: An upward-opening graph has a minimum. A downward-opening graph has a maximum.

Giving an intercept as only one coordinate when the full point is needed.

Correction: An x-intercept with value

3

is

(3,0)

. A y-intercept with value

6

is

(0,6)

.

Lesson summary

In $y=a(x-h)^2+k$ , the vertex is $(h,k)$ and the axis is $x=h$ .
For $y=ax^2+bx+c$ , the axis is $x=-\frac{b}{2a}$ .
Set $y=0$ for x-intercepts and $x=0$ for the y-intercept.
The domain of a quadratic function is all real numbers.
The vertex and opening direction determine the range.

Check your understanding

Question 1

For

y=(x+2)^2-5

, what is the vertex?

$(-2,-5)$
$(2,-5)$
$(-2,5)$
$(0,-5)$

Show answer and explanation

(-2,-5)

The bracket is zero when

x=-2

, and the value outside the bracket gives the vertex’s vertical coordinate,

-5

.

Question 2

A quadratic opens downward and has vertex

(3,7)

. Which statement describes its range?

$y\ge 7$
$y\le 7$
All real numbers
$x\le 7$

Show answer and explanation

y\le 7

The vertex is the maximum point, so outputs cannot be greater than

7

.

Question 3

What is the axis of symmetry for

y=x^2-6x+5

?

$x=-3$
$x=3$
$y=3$
$x=6$

Show answer and explanation

x=3

Here

a=1

and

b=-6

. Substitution into

x=-\frac{b}{2a}

gives

x=3

.

Key terms

Quadratic function: A function that can be written as $y=ax^2+bx+c$ , where $a\ne 0$ .
Vertex: The turning point of a quadratic graph; it is the graph’s minimum or maximum.
Axis of symmetry: The vertical line that divides a quadratic graph into matching left and right sides.
Intercept: A point where a graph meets one of the coordinate axes.
Domain: The set of input values, or $x$ -values, for which a function is defined.
Range: The set of output values, or $y$ -values, that a function can have.

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Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation RF1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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RF1 · Quadratic Graphs: Vertex, Intercepts and Range

Alberta Grade 11 Mathematics

Vertices, intercepts, domain, range, and symmetry

What you will learn

1. Prerequisite review: coordinates and intercepts

2. Vertex, symmetry, domain, and range

3. Connect equations to a calculated graph

4. Student try and reasonableness checks

Read features from vertex form

Find features from expanded form

Use intercepts to locate the axis

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 20-2

Tutor support for RF1 · Describe quadratic vertices, intercepts, domain, range, and symmetry

About this lesson and its review