Parent Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parent 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 parent function is the simplest, most basic version of a function family — the unshifted, unstretched, unreflected template. All other functions in the family are transformations of this parent. Memorizing parent function shapes allows rapid graphing of transformed versions.

It is the original template shape you move, stretch, or reflect.

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 parent function is the simplest, untransformed member of a family — everything else is a shift, stretch, or flip of it.

Common stuck point: The procedure for parent functions is the easy part; the trap is treating a shifted/stretched graph as a new family. Asking "Is this the simplest untransformed template that all others in the family are built from?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this the simplest untransformed template that all others in the family are built from?

Worked Examples

Example 1

easy

Identify the parent function of

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

and describe the transformations.

Answer

\text{Parent: } y = x^2; \text{ right 2, vertical stretch by 3, up 5}

First step

The parent function is

y = x^2

(the basic quadratic function).

Full solution

2
The transformation $(x-2)$ : horizontal shift $2$ units to the right.
3
The coefficient $3$ : vertical stretch by a factor of $3$ .
4
The $+5$ : vertical shift $5$ units up. The vertex moves from $(0, 0)$ to $(2, 5)$ .

A parent function is the simplest form of a function family. Common parent functions include

y = x

(linear),

y = x^2

(quadratic),

y = |x|

(absolute value),

y = \sqrt{x}

(square root), and

y = 1/x

(reciprocal). All other functions in the family are transformations of the parent.

Example 2

medium

Match each equation to its parent function: (a)

y = -\sqrt{x+3}

, (b)

y = \frac{2}{x-1}

, (c)

y = |x| - 4

, (d)

y = 2^{x+1} - 3

Example 3

medium

Identify the parent of

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

and list transformations in order.

Example 4

medium

The graph of

y = 2|x - 1| + 5

has what vertex?

Example 5

hard

Write a function whose parent is

y = \frac{1}{x}

with vertical asymptote

x = 4

and horizontal asymptote

y = -1

, passing through

(5, 1)

Example 6

hard

Parent function

y = e^x

is shifted left

2

, stretched vertically by factor

3

, and reflected over the

x

-axis. Write the new equation.

Example 7

challenge

Given parent

y = \sqrt{x}

, write a transformation with domain

[-3, \infty)

, range

(-\infty, 4]

, passing through

(-3, 4)

Practice Problems

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

Example 1

medium

Sketch the key features (domain, range, intercepts, asymptotes) of the parent function

y = \frac{1}{x}

Example 2

hard

Write a single equation using transformations of the parent function

y = \sqrt{x}

that has domain

(-\infty, 4]

, range

[-2, \infty)

, and passes through the point

(0, 0)

Example 3

easy

Name the parent function of

y = (x-3)^2 + 1

Example 4

easy

Name the parent function of

y = |x| + 4

Example 5

easy

Name the parent function of

y = \sqrt{x} - 2

Example 6

easy

Name the parent function of

y = 2^x + 3

Example 7

easy

Name the parent function of

y = 5x - 7

Example 8

easy

What is the domain of the parent function

y = \sqrt{x}

Example 9

easy

Name the parent function of

y = \frac{1}{x} + 6

Example 10

easy

Name the parent function of

y = x^3 - 5

Example 11

medium

In what direction does

y = (x+4)^2

shift the parent

y=x^2

Example 12

medium

Describe the transformation from

y=x^2

y=(x-3)^2+2

Example 13

medium

What transformation turns

y=|x|

into

y=-|x|

Example 14

medium

Name the parent function and its range for

y = -3\cdot 2^x

Example 15

medium

The point

(0,0)

is the vertex of

y=x^2

. Where is the vertex of

y=(x-2)^2+5

Example 16

medium

Which parent function is symmetric about the

y

-axis:

y=x^2

y=x^3

Example 17

medium

Name the parent function of

y = 3\ln(x) - 1

Example 18

medium

State the domain of the parent

y = \frac{1}{x}

Example 19

medium

Describe the transformation from

y=\sqrt{x}

y=\sqrt{x}+3

Example 20

challenge

y = -2\sqrt{x-1} + 3

comes from which parent, and list every transformation in order.

Example 21

challenge

A parent function passes through

(0,1)

and grows without bound as

x\to\infty

but approaches

0

x\to-\infty

. Name it.

Example 22

challenge

y = f(2x)

is applied to the parent

y=\sqrt{x}

. Does the graph stretch or compress horizontally, and by what factor?

Example 23

easy

Name the parent function of

y = (x+5)^2 - 7

Example 24

easy

What is the range of the parent function

y = x^2

Example 25

easy

Identify the parent of

y = -|x-2|

Example 26

easy

Identify the parent of

y = -\frac{1}{x} + 4

Example 27

medium

What transformations turn

y = x^2

into

y = (x-4)^2 - 9

Example 28

medium

What transformation maps

y = \sqrt{x}

y = \sqrt{-x}

Example 29

medium

State the asymptotes of

y = \frac{1}{x - 3} + 2

Example 30

medium

What is the parent function of

y = \sqrt[3]{x - 2}

Example 31

medium

Compare end behavior: which is faster as

x \to \infty

— parent

y = 2^x

y = x^2

Example 32

hard

Given

y = a\sqrt{x - h} + k

passes through

(4, 1)

and

(9, 4)

with

h = 0

, find

a

and

k

Example 33

hard

Describe the transformation from

y = x^2

y = (2x - 6)^2

Example 34

hard

A parent function passes through

(1, 0)

and approaches

-\infty

x \to 0^+

. Name it.

Example 35

challenge

For the function

y = -3 \cdot 2^{-(x-1)} + 4

, identify the parent and end behavior as

x \to \infty

Example 36

challenge

For the parent

y = \frac{1}{x}

, prove the graph is symmetric about the line

y = x

← Back to Parent Functions Practice Mode

Related Concepts

Function Families Function Transformation Multiple Representations

Background Knowledge

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

function familiestransformationmultiple representations