Quadratic Vertex Form Examples in Math

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

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

Concept Recap

A quadratic written as $f(x) = a(x - h)^2 + k$ , where the vertex $(h, k)$ is directly readable from the formula.

Imagine sliding a basic $x^2$ parabola around on the coordinate plane. The value $h$ shifts it left or right, $k$ shifts it up or down, and $a$ stretches or flips it. The vertex $(h, k)$ is the parabola's turning point—you can read it directly from this form.

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: Vertex form $a(x-h)^2+k$ puts the parabola's vertex $(h,k)$ right in the formula.

Common stuck point: The procedure for quadratic vertex form is the easy part; the trap is taking $h$ with the wrong sign. Asking "Is the quadratic written as a squared binomial plus a constant, and do I want its turning point?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the quadratic written as a squared binomial plus a constant, and do I want its turning point?

Worked Examples

Example 1

easy

What is the vertex of

f(x) = 2(x - 3)^2 + 1

Answer

(3, 1)

First step

Vertex form is

a(x - h)^2 + k

with vertex

(h, k)

Full solution

2
Here $h = 3$ and $k = 1$ .
3
The vertex is $(3, 1)$ .

In vertex form, the vertex coordinates are read directly:

h

is the value subtracted from

x

, and

k

is the constant added.

Example 2

medium

Write the vertex form of a parabola with vertex

(-1, 4)

passing through

(0, 7)

Example 3

easy

Write the vertex form of a parabola with vertex

(2, -3)

and

a = 1

Example 4

medium

Write the vertex form of a parabola with vertex

(3, 2)

passing through

(5, 10)

Example 5

medium

Convert

f(x) = x^2 + 6x + 13

to vertex form.

Example 6

medium

Convert

f(x) = 2x^2 - 12x + 7

to vertex form.

Example 7

medium

Translate the parabola

y = x^2

right

3

units and down

4

units. Write the vertex form.

Example 8

hard

Convert

y = 3x^2 + 12x + 7

to vertex form.

Example 9

hard

Write the vertex form of a parabola with vertex

(-2, 5)

that passes through

(0, 1)

Example 10

hard

Convert

y = \tfrac{1}{2}x^2 - 3x + 4

to vertex form.

Example 11

challenge

A parabola passes through

(1,3)

(2,2)

, and

(4,6)

. Write it in vertex form.

Practice Problems

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

Example 1

easy

What is the vertex of

g(x) = -(x + 5)^2 - 2

Example 2

hard

Convert

x^2 + 4x + 7

to vertex form.

Example 3

easy

Find the vertex of

y = (x - 3)^2 + 4

Example 4

easy

Find the vertex of

y = (x + 2)^2 - 5

Example 5

easy

Does

y = (x-1)^2 + 3

open up or down?

Example 6

easy

What is the minimum value of

y = (x-2)^2 + 7

Example 7

easy

Find the axis of symmetry of

y = (x+4)^2 + 1

Example 8

easy

What is the maximum value of

y = -(x-1)^2 + 6

Example 9

easy

Find the vertex of

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

medium

A parabola has vertex

(2, -1)

and passes through

(0, 3)

. Find

a

in vertex form.

Example 16

medium

Write the vertex form of a parabola with vertex

(-3, 4)

and

a = 2

Example 17

medium

Convert

y = -x^2 + 4x - 1

to vertex form.

Example 18

medium

Convert

y = x^2 + 2x - 3

to vertex form.

Example 19

medium

A parabola has vertex

(0, 5)

and passes through

(2, 1)

. Find its vertex-form equation.

Example 20

challenge

A ball's height is

h = -16t^2 + 64t

. Write in vertex form and give the maximum height.

Example 21

challenge

Two parabolas

y=(x-h)^2+k

both pass through

(1,0)

and

(5,0)

. Find

h

Example 22

challenge

Find

k

so that

y = (x-2)^2 + k

is tangent to the

x

-axis.

Example 23

easy

What is the vertex of

g(x) = -3(x+1)^2 + 6

Example 24

easy

Find the minimum value of

y = (x-4)^2 + 9

Example 25

easy

Find the axis of symmetry of

y = (x-6)^2 - 3

Example 26

medium

Convert

f(x) = x^2 - 8x + 11

to vertex form by completing the square.

Example 27

medium

Find the vertex of

f(x) = 2(x-1)^2 - 8

and its zeros.

Example 28

medium

Write the vertex form of a parabola with vertex

(0, -2)

and

y

-intercept

-2

passing through

(3, 16)

Example 29

medium

Find the vertex of

y = -(x-2)^2

Example 30

hard

A ball's height in metres is

h(t) = -5(t-2)^2 + 25

where

t

is seconds. What is the maximum height and when?

Example 31

hard

Find the vertex of

f(x) = -2x^2 - 8x - 3

using vertex form.

Example 32

hard

For what value(s) of

k

does

y = (x-3)^2 + k

have exactly one real zero?

← Back to Quadratic Vertex Form Formula Explained Practice Mode

Related Concepts

Quadratic Functions Quadratic Standard Form Completing the Square Graphing Parabolas Vertex and Axis of Symmetry

Background Knowledge

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

quadratic functionsquadratic standard form