Graphing Parabolas Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Graphing Parabolas.

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 process of plotting a quadratic function by identifying its key features: vertex, axis of symmetry, direction of opening, $y$ -intercept, and $x$ -intercepts (if they exist).

A parabola is a U-shaped curve (or upside-down U). Start by finding the vertex—that is the turning point. Then the axis of symmetry tells you the curve is a mirror image on both sides. Plot a few symmetric points and connect them in a smooth curve.

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: Graphing a parabola means plotting its vertex, axis, intercepts, and symmetric points into a smooth U.

Common stuck point: The procedure for graphing parabolas is the easy part; the trap is getting the opening direction wrong. Asking "Am I producing or reading a picture of a quadratic, using vertex, axis, and intercepts?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I producing or reading a picture of a quadratic, using vertex, axis, and intercepts?

Worked Examples

Example 1

easy

Identify the key features of

f(x) = x^2 - 4x + 3

for graphing.

Key features of f(x) = x² − 4x + 3

Answer

Vertex

(2,-1)

, opens up,

x

-intercepts at 1 and 3,

y

-intercept at 3.

First step

Direction:

a = 1 > 0

, opens upward.

Full solution

2
Vertex: $x = -\frac{-4}{2} = 2$ ; $f(2) = 4 - 8 + 3 = -1$ . Vertex: $(2, -1)$ .
3
$y$ -intercept: $f(0) = 3$ , point $(0, 3)$ .
4
$x$ -intercepts: factor $x^2 - 4x + 3 = (x-1)(x-3) = 0$ , so $x = 1$ and $x = 3$ .

To graph a parabola, find: (1) direction from the sign of

a

, (2) vertex, (3)

y

-intercept at

x=0

, (4)

x

-intercepts by setting

f(x) = 0

Example 2

medium

Sketch

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

. Find vertex and intercepts.

Key features of g(x) = −x² + 2x + 3

Example 3

easy

Find the vertex and

y

-intercept of

y = -x^2 + 4x

Vertex and y-intercept of y = −x² + 4x

Example 4

medium

For

y = x^2 - 2x - 8

, find vertex, axis,

y

-intercept, and

x

-intercepts.

All key features of y = x² − 2x − 8

Example 5

medium

A parabola has vertex

(2, -5)

and passes through

(0, -1)

. Write it in vertex form.

Example 6

medium

For

y = -x^2 + 6x - 5

, find all key features for graphing.

Key features of y = −x² + 6x − 5

Example 7

medium

Find the range of

y = -2x^2 + 8x - 3

Example 8

hard

A parabola has

x

-intercepts

-2

and

6

and passes through

(0, -6)

. Find its equation.

Example 9

hard

Sketch the key features of

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

and find its

x

-intercepts.

Key features of y = −2(x−1)² + 8

Example 10

hard

The minimum of

y = x^2 + bx + 9

0

. Find

b

Example 11

challenge

A parabola has

x

-intercepts

-1

and

5

, and its maximum value is

9

. Find its equation in standard form.

Parabola through x-intercepts −1 and 5 with maximum 9

Practice Problems

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

Example 1

easy

Does

f(x) = 5x^2 + 1

have any

x

-intercepts?

Example 2

medium

The vertex of a parabola is

(0, -4)

and it passes through

(2, 0)

. Find the equation.

Example 3

easy

Does

y = x^2 - 3

open up or down?

Example 4

easy

Find the

y

-intercept of

y = x^2 + 2x + 5

Example 5

easy

Find the axis of symmetry of

y = x^2 - 6x + 1

Example 6

easy

Find the vertex of

y = x^2 - 4x + 3

Find the vertex of y = x² − 4x + 3

Example 7

easy

How many

x

-intercepts does

y = x^2 + 4

have?

Example 8

easy

The vertex of

y = (x-1)^2 + 2

is a minimum or maximum?

Example 9

easy

Find the

y

-intercept of

y = 2x^2 - x + 7

Example 10

easy

If a parabola has vertex

(3,2)

, what is its axis of symmetry?

Example 11

medium

Find the vertex and direction of

y = -2x^2 + 8x - 5

Vertex and direction of y = −2x² + 8x − 5

Example 12

medium

Find the

x

-intercepts of

y = x^2 - 5x + 6

Example 13

medium

Sketch info for

y = x^2 - 2x - 3

: vertex,

y

-intercept,

x

-intercepts.

Key features of y = x² − 2x − 3

Example 14

medium

Use symmetry: if

y=x^2-4x+1

passes through

(0,1)

, find the mirror point.

Example 15

medium

Does

y = x^2 + 2x + 5

cross the

x

-axis? Use the discriminant.

Example 16

medium

A downward parabola has vertex

(2, 9)

and a root at

x=5

. Find the other root.

By symmetry, find the other root of the downward parabola

Example 17

medium

Find where

y = x^2 - 4

and

y = 0

meet, then sketch direction.

y = x² − 4 meeting y = 0

Example 18

medium

Find the vertex of

y = x^2 + 2x - 8

and its

y

-intercept.

Example 19

medium

Find the

x

-intercepts of

y = 2x^2 - 8

Example 20

challenge

A parabola has axis of symmetry

x = 1

and passes through

(3, 0)

and

(0, 6)

. Find its equation in factored form.

Example 21

challenge

For what values of

c

does

y = x^2 - 6x + c

have its vertex on the

x

-axis?

Example 22

challenge

The graph of

y=ax^2

passes through

(2, 12)

. Find

a

and the

y

-value at

x=-2

Example 23

easy

Find the

y

-intercept of

y = x^2 - 3x + 8

Example 24

easy

Find the axis of symmetry of

y = x^2 + 8x + 1

Example 25

easy

Find the vertex of

y = x^2 - 6x + 5

Find the vertex of y = x² − 6x + 5

Example 26

easy

Find the

x

-intercepts of

y = x^2 - 9

Example 27

easy

What is the axis of symmetry of

y = (x+3)^2 - 7

Example 28

medium

Find the

x

-intercepts of

y = 2x^2 + 5x - 3

Example 29

medium

Find the vertex of

y = 2x^2 - 8x + 1

Example 30

medium

By symmetry, if

(1, 7)

lies on a parabola with axis

x = 4

, what other point on the parabola has the same

y

-value?

Example 31

hard

Find the vertex of

y = 3x^2 + 12x + 5

and state max or min value.

Example 32

hard

For what values of

c

does

y = x^2 - 4x + c

have two distinct

x

-intercepts?

Example 33

hard

Where do

y = x^2

and

y = x + 2

intersect?

Intersections of y = x² and y = x + 2

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Related Concepts

Quadratic Vertex Form Coordinate Plane Vertex and Axis of Symmetry Function Transformation

Background Knowledge

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

quadratic vertex formcoordinate plane