Vertex and Axis of Symmetry Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Vertex and Axis of Symmetry.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The vertex of a parabola is the point where it reaches its maximum or minimum value. The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.

Fold the parabola along the axis of symmetry and both halves match perfectly. The vertex is at the fold—the very bottom of a U-shaped parabola or the very top of an upside-down one. It is the point where the function changes direction.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The axis of symmetry is the mirror line $x=-\tfrac{b}{2a}$ ; the vertex is the point on it where the parabola turns.

Common stuck point: The procedure for vertex and axis of symmetry is the easy part; the trap is reporting only the x-coordinate as 'the vertex'. Asking "Am I after the turning point or the mirror line of a parabola?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I after the turning point or the mirror line of a parabola?

Worked Examples

Example 1

easy

Find the vertex and axis of symmetry of

f(x) = x^2 + 8x + 12

Answer

Vertex

(-4, -4)

; axis

x = -4

First step

Axis of symmetry:

x = -\frac{b}{2a} = -\frac{8}{2} = -4

Full solution

2
Vertex $y$ -value: $f(-4) = 16 - 32 + 12 = -4$ .
3
Vertex: $(-4, -4)$ ; axis of symmetry: $x = -4$ .

The axis of symmetry is the vertical line

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

that passes through the vertex and divides the parabola into two mirror-image halves.

Example 2

medium

f(2) = 7

and the axis of symmetry is

x = 5

, find

f(8)

Example 3

medium

A ball's height in feet is

h(t) = -16t^2 + 64t + 5

. When does it reach maximum height, and how high?

Example 4

hard

Show why the vertex of

y = ax^2 + bx + c

has

y

-coordinate

c - \frac{b^2}{4a}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the axis of symmetry of

y = 3x^2 - 12x + 5

Example 2

medium

Is the vertex of

y = -2(x-3)^2 + 10

a maximum or minimum?

Example 3

easy

Find the axis of symmetry of

y = x^2 + 8x + 1

Example 4

easy

Find the vertex

x

-coordinate of

y = 2x^2 - 12x + 5

Example 5

easy

Read the vertex of

y = (x-6)^2 + 2

Example 6

easy

Is the axis of symmetry a line or a point?

Example 7

easy

Find the vertex of

y = x^2 - 2x + 1

Example 8

easy

What is the axis of symmetry of

y = -3(x+5)^2 + 1

Example 9

easy

The vertex of an upward parabola represents a ___?

Example 10

easy

Find the vertex of

y = x^2 + 4x

Example 11

medium

Find the vertex of

y = 3x^2 - 6x + 4

Example 12

medium

Find the vertex of

y = -x^2 + 6x - 5

and state min or max.

Example 13

medium

Convert

y = x^2 + 10x + 30

to vertex form to find the vertex.

Example 14

medium

Two points

(1, 4)

and

(7, 4)

lie on a parabola. Find the axis of symmetry.

Example 15

medium

A parabola has axis

x=2

and passes through

(5, 0)

. Where is the other

x

-intercept?

Example 16

medium

Find the maximum value of

y = -2x^2 + 8x + 1

Example 17

medium

Where is the vertex of

y = (x-3)^2 - (x-3)

? (Hint: let

u=x-3

Example 18

medium

Find the vertex of

y = x^2 + 6x + 5

Example 19

medium

The points

(-1, 3)

and

(5, 3)

lie on a parabola. Find the vertex

x

-coordinate.

Example 20

challenge

Show that the vertex of

y = ax^2 + bx + c

has

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

using symmetry of the roots.

Example 21

challenge

A parabola has minimum value

-4

x=1

and

y

-intercept

-3

. Find

a

Example 22

challenge

For

y = x^2 + bx + 7

, the axis of symmetry is

x=3

. Find

b

and the minimum value.

Example 23

easy

Find the axis of symmetry of

y = x^2 - 10x + 7

Example 24

easy

Find the vertex of

y = (x+3)^2 - 7

Example 25

easy

Find the vertex of

y = -(x-4)^2 + 9

Example 26

easy

Find the axis of symmetry of

y = 4x^2 + 16x - 1

Example 27

easy

What is the axis of symmetry of

y = x^2

Example 28

easy

Find the vertex of

y = x^2 - 6x + 10

Example 29

medium

Convert

y = x^2 + 8x + 19

to vertex form.

Example 30

medium

Find the vertex of

y = 2x^2 - 8x + 11

Example 31

medium

Find the maximum value of

y = -x^2 + 4x + 5

Example 32

medium

A parabola has

x

-intercepts

-2

and

8

. Find the axis of symmetry.

Example 33

medium

f(x) = ax^2 + bx + c

has

f(1) = f(7)

, find the axis of symmetry.

Example 34

medium

A parabola has axis

x=3

and passes through

(0, 7)

. Find the value at

x = 6

Example 35

medium

Find the vertex of

y = -3x^2 + 12x - 5

Example 36

medium

What is the minimum value of

y = x^2 + 6x + 13

Example 37

medium

If the vertex of

y = x^2 + bx + c

(4, -3)

, find

b

and

c

Example 38

hard

A rectangle's perimeter is

40

. Express its area as a quadratic in the width

w

, then find the maximum area.

Example 39

hard

Find the vertex of

y = 5(x-2)^2 + 7

and state the axis of symmetry.

Example 40

hard

f(x) = 2x^2 + bx + 1

has vertex on the line

y = -7

, find all possible

b

Example 41

hard

A parabola passes through

(1, 0)

(5, 0)

, and

(3, 8)

. Find its equation in vertex form.

Example 42

hard

The graph of

y = x^2 + px + q

is tangent to the x-axis. Express

q

in terms of

p

Example 43

challenge

A farmer has

120

m of fence to enclose a rectangular pen with one side along a river (no fence needed on that side). Find dimensions that maximize area.

Example 44

challenge

For what value(s) of

k

does the parabola

y = x^2 + kx + 4

have its vertex on the line

y = x

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Related Concepts

Quadratic Functions Symmetry Graphing Parabolas Optimization

Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic functionssymmetry