Quadratic Functions Examples in Math

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

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 function is a polynomial function of degree 2, written as $f(x) = ax^2 + bx + c$ with $a \neq 0$ , whose graph is a U-shaped curve called a parabola that opens upward when $a > 0$ or downward when $a < 0$ .

The path of a thrown ball — rising then falling — traces a parabola opening downward.

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: A quadratic function $f(x)=ax^2+bx+c$ graphs as a U-shaped parabola opening up if $a>0$ , down if $a<0$ .

Common stuck point: The procedure for quadratic functions is the easy part; the trap is forgetting $a$ may be negative. Asking "Is the highest power of the variable exactly 2, so the graph curves into a parabola?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the highest power of the variable exactly 2, so the graph curves into a parabola?

Worked Examples

Example 1

easy

Find the vertex of

f(x) = x^2 - 6x + 8

Answer

(3, -1)

First step

The

x

-coordinate of the vertex is

x = -\frac{b}{2a} = -\frac{-6}{2(1)} = 3

Full solution

2
The $y$ -coordinate is $f(3) = 9 - 18 + 8 = -1$ .
3
The vertex is $(3, -1)$ .

For a quadratic

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

, the vertex formula

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

gives the axis of symmetry. Plugging this

x

back in gives the minimum (if

a > 0

) or maximum (if

a < 0

) value.

Example 2

medium

Does the parabola

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

open upward or downward? Find its maximum value.

Example 3

medium

Find the vertex and axis of symmetry of

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

Example 4

medium

Convert

f(x) = x^2 + 6x + 4

to vertex form by completing the square.

Example 5

medium

A farmer has

80

m of fencing for a rectangular plot against a wall (so only three sides need fencing). What dimensions maximize the area?

Example 6

hard

Find the equation of the tangent line to

y = x^2

at the point

(3, 9)

using only quadratic properties (without calculus).

Practice Problems

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

Example 1

easy

Find the

y

-intercept of

f(x) = 3x^2 - 2x + 5

Example 2

medium

Find the zeros of

f(x) = x^2 - 4x - 5

Example 3

easy

f(x) = x^2 - 4

a quadratic function?

Example 4

easy

Which way does

y = -2x^2 + 3

open?

Example 5

easy

Evaluate

f(x) = x^2 + 1

x = 3

Example 6

easy

What is the constant term

c

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

Example 7

easy

Find

f(0)

for

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

Example 8

easy

How many

x

-intercepts can a parabola have at most?

Example 9

easy

Is the vertex of

y = x^2

a maximum or minimum?

Example 10

easy

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

, why must

a \ne 0

Example 11

medium

Find the axis of symmetry of

y = x^2 - 6x + 5

Example 12

medium

Find the vertex of

y = x^2 - 6x + 5

Example 13

medium

Find the zeros of

f(x) = x^2 - 5x + 6

by factoring.

Example 14

medium

Use the discriminant to count real roots of

x^2 + x + 1 = 0

Example 15

medium

Find the minimum value of

f(x) = x^2 - 4x + 7

Example 16

medium

Convert

y = (x - 3)^2 + 2

to identify the vertex.

Example 17

medium

A ball's height is

h(t) = -16t^2 + 32t

. When does it return to the ground?

Example 18

challenge

Solve

2x^2 - 3x - 2 = 0

with the quadratic formula.

Example 19

challenge

For what

k

does

x^2 + kx + 9 = 0

have exactly one real root?

Example 20

challenge

The product of a quadratic's roots is

6

and their sum is

5

. Find the quadratic (monic).

Example 21

medium

How many real zeros does

f(x) = x^2 + 4

have?

Example 22

medium

Write

f(x) = x^2 - 9

in factored form.

Example 23

easy

Which way does the parabola

f(x) = 4x^2 - 1

open?

Example 24

easy

Find

f(-2)

for

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

Example 25

easy

Find the vertex of

f(x) = x^2 + 4x + 1

Example 26

easy

Write

f(x) = (x-2)^2 - 5

in standard form.

Example 27

medium

Find the range of

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

Example 28

medium

Find the

x

-intercepts of

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

Example 29

medium

A parabola has vertex

(2, -5)

and passes through

(0, -1)

. Find its equation in vertex form.

Example 30

medium

Find the maximum value of

g(x) = -3x^2 + 12x - 7

Example 31

medium

Describe the transformation from

y = x^2

y = (x + 1)^2 - 4

Example 32

medium

A quadratic has zeros at

x = -2

and

x = 5

and passes through

(0, -20)

. Find

f(x)

Example 33

hard

Find the values of

x

for which

f(x) = x^2 - 6x + 5 > 0

Example 34

hard

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

has vertex

(1, 4)

and

f(3) = 0

, find

a

b

c

Example 35

hard

The graph of

f(x) = 2x^2 + bx - 6

has

x

-intercept at

x = 3

. Find

b

and the other

x

-intercept.

Example 36

hard

A rocket's height (m) at time

t

(s) is

h(t) = -5t^2 + 40t

. What is its maximum height and when does it occur?

Example 37

hard

Find the equation of the parabola with

x

-intercepts

-3

and

1

and passing through

(2, 10)

Example 38

hard

For what values of

k

does

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

have two distinct real zeros?

Example 39

hard

The parabola

y = ax^2 + bx + c

passes through

(0, 1)

(1, 0)

(2, 1)

. Find

a, b, c

Example 40

challenge

f(x) = x^2 + px + q

and

f(x) \geq 0

for all real

x

, what is the relation between

p

and

q

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Related Concepts

Linear Functions Exponents Quadratic Formula Polynomials

Background Knowledge

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

linear functionsexponents