Practice Function Families in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

A function family is a group of functions sharing the same general form and behavior, differing only in the values of one or more parameters.

$y = mx + b$ is a family of lines. Different $m$ and $b$ give different lines.

Showing a random 20 of 50 problems.

Example 1

medium

Two parabolas are

y = x^2

and

y = -2x^2

. Name their shared family and two ways the parameter changes the graph.

Example 2

hard

Classify

y = e^{-x^2}

— is it linear, polynomial, exponential, or something else?

Example 3

challenge

A family is defined by

f(x) = a x^2

with

a \ne 0

. Prove that every member passes through the origin and explain why varying

a

cannot change that.

Example 4

easy

y = a x^2 + b x + c

, what role does

a

play (besides being a parameter)?

Example 5

medium

A table shows

f(1)=2

f(2)=5

f(3)=10

f(4)=17

. Identify the family.

Example 6

medium

What family is

y = \sqrt[3]{x}

(cube root)?

Example 7

hard

A linear and an exponential function both pass through

(0, 5)

and

(1, 10)

. Find each. At

x = 4

, which is larger?

Example 8

challenge

A mystery function has these values:

f(1) = 2

f(2) = 6

f(3) = 18

f(4) = 54

. Identify the family and the specific member.

Example 9

hard

For the family

y = a \sin(bx) + c

, which parameter changes the vertical midline?

Example 10

hard

The family

y=x^n

for integer

n\geq1

has different behavior depending on whether

n

is even or odd. Classify and explain the symmetry of

y=x^2

y=x^3

y=x^4

y=x^5

Example 11

medium

Classify

y = 4^x

versus

y = x^4

by family, and state which grows faster for large

x

Example 12

medium

The family

y = \frac{k}{x}

(with

k \neq 0

). Describe the role of

k

on the graph.

Example 13

medium

Which family does

y = 5\sin(2x)

belong to?

Example 14

medium

Two members of the linear family pass through given points: line A has slope 2, line B has slope 2 but a different intercept. What is true about their graphs?

Example 15

easy

y = \frac{1}{x}

in the same family as

y = x^2

Example 16

easy

Classify

y = -4x + 1

by family.

Example 17

medium

A function doubles every time

x

increases by 1, and equals 5 at

x = 0

. Identify its family and write the specific member.

Example 18

easy

In the family

y = a(x - h)^2 + k

, which parameter shifts the vertex horizontally?

Example 19

hard

A radioactive sample has half-life

5

years. Identify the family and write a model starting from

80

Example 20

medium

A line and a parabola both pass through

(0, 0)

and

(2, 4)

. Identify each by family.

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Related Concepts

Parameter Function Parent Functions Function Transformation