Practice Parent Functions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

challenge

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

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

Example 2

easy

What is the domain of the parent function

y = \sqrt{x}

Example 3

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 4

medium

Sketch the parent

y = x^3

: list its key features (domain, range, symmetry, intercepts).

Example 5

easy

Name the parent function of

y = x^3 - 5

Example 6

medium

State the domain of the parent

y = \frac{1}{x}

Example 7

medium

Which parent function is symmetric about the

y

-axis:

y=x^2

y=x^3

Example 8

medium

Describe the transformation from

y=x^2

y=(x-3)^2+2

Example 9

challenge

For the function

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

, identify the parent and end behavior as

x \to \infty

Example 10

medium

In what direction does

y = (x+4)^2

shift the parent

y=x^2

Example 11

medium

Describe the transformation from

y=\sqrt{x}

y=\sqrt{x}+3

Example 12

challenge

y = f(2x)

is applied to the parent

y=\sqrt{x}

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

Example 13

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 14

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 15

easy

Identify the parent function of

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

and describe the transformations.

Example 16

medium

The point

(0,0)

is the vertex of

y=x^2

. Where is the vertex of

y=(x-2)^2+5

Example 17

easy

Name the parent function of

y = (x-3)^2 + 1

Example 18

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 19

easy

Identify the parent of

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

Example 20

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.

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Related Concepts

Function Families Function Transformation Multiple Representations