Function Families Examples in Math

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

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 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.

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 function family is all functions sharing one general form, distinguished only by their parameter values.

Common stuck point: The procedure for function families is the easy part; the trap is treating $m$ and $b$ as variables. Asking "Do these functions all share one general form, differing only in the values of their constants?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do these functions all share one general form, differing only in the values of their constants?

Worked Examples

Example 1

easy

The family of quadratics

f(x)=ax^2

(with

a\neq0

) all share the same vertex at the origin. Describe how changing

a

from

1

4

-2

affects the graph.

Answer

a>0

: opens up;

a<0

: opens down;

|a|>1

: narrower;

|a|<1

: wider

First step

a=1

: standard parabola, opens up, vertex at origin.

f(3)=9

Full solution

2
$a=4$ : opens up, narrower (vertically stretched by $4$ ). $f(3)=36$ .
3
$a=-2$ : opens down (reflected), vertically stretched by $2$ . $f(3)=-18$ . All share vertex $(0,0)$ and $x$ -intercept at $x=0$ . Parameter $a$ controls direction and width.

A function family is a set of functions sharing a common form, differing only in parameter values. Varying

a

ax^2

produces every possible parabola with vertex at the origin, illustrating how one parameter controls an entire continuum of shapes.

Example 2

medium

The family

f_k(x)=\dfrac{1}{x-k}

has a vertical asymptote at

x=k

for each parameter

k

. Analyze the graphs for

k=-2, 0, 3

and describe the pattern.

Example 3

medium

A function passes through

(0, 4)

and triples every time

x

increases by

1

. Identify the family and find the specific member.

Example 4

medium

The family

y = \frac{k}{x}

(with

k \neq 0

). Describe the role of

k

on the graph.

Example 5

hard

Classify

y = e^{-x^2}

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

Practice Problems

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

Example 1

easy

In the family

y=b^x

(exponential), how does the graph change as

b

varies? Compare

b=2

b=1

b=0.5

x=2

Example 2

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 3

easy

To which family does

y = 3x + 7

belong: linear, quadratic, or exponential?

Example 4

easy

To which family does

y = 3 \cdot 2^x

belong?

Example 5

easy

To which family does

y = x^2 - 5x + 6

belong?

Example 6

easy

In the family

y = mx + b

, which letters are parameters and which is the variable?

Example 7

easy

y = x^2

and

y = x^2 + 5

belong to the same family?

Example 8

easy

Which family has graphs that are all straight lines?

Example 9

easy

y = \frac{1}{x}

in the same family as

y = x^2

Example 10

easy

y = a x^2 + b x + c

, what role does

a

play (besides being a parameter)?

Example 11

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 12

medium

A table shows constant first differences of 4 between successive

y

-values at equal

x

-steps. Which family fits, and what does the 4 represent?

Example 13

medium

A table shows constant SECOND differences (first differences are

3, 5, 7

). Which family fits?

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

medium

Classify

y = 4^x

versus

y = x^4

by family, and state which grows faster for large

x

Example 16

medium

The general member of a family is

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

. What does this family represent, and what do

h

and

k

control?

Example 17

medium

Given that all members of a family satisfy

f(x) = a \cdot b^x

with

b > 0

, what feature do ALL members share regardless of

a

and

b

Example 18

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 19

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 20

challenge

Distinguish the families of

y = 2^x

y = x^2

, and

y = 2x

by comparing

f(x+1)

f(x)

for each (additive vs multiplicative step behavior).

Example 21

medium

Two parabolas are

y = x^2

and

y = -2x^2

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

Example 22

medium

Classify by family and growth: which is larger for large

x

, the linear

y = 100x

or the quadratic

y = x^2

Example 23

easy

Classify

y = -4x + 1

by family.

Example 24

easy

Classify

y = 7 \cdot \left(\tfrac{1}{2}\right)^x

by family.

Example 25

easy

Classify

y = |x - 3| + 1

by family.

Example 26

easy

Classify

y = \log_2(x)

by family.

Example 27

easy

In the family

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

, which parameter shifts the vertex horizontally?

Example 28

easy

Identify the family:

y = \frac{1}{x}

Example 29

medium

A table shows

f(1)=3

f(2)=5

f(3)=7

f(4)=9

. Identify the family and the specific member.

Example 30

medium

A table shows

f(1)=2

f(2)=5

f(3)=10

f(4)=17

. Identify the family.

Example 31

medium

Compare the end behavior as

x \to \infty

for

y = x^3

and

y = 2^x

. Which grows faster?

Example 32

medium

The family

y = a \cdot b^x

with

a > 0

and

b > 1

. What is true of every member at

x = 0

Example 33

medium

A line and a parabola both pass through

(0, 0)

and

(2, 4)

. Identify each by family.

Example 34

medium

y = a \cdot b^x

, what point do all members share?

Example 35

hard

A mystery function has

f(0)=3

f(1)=12

f(2)=48

f(3)=192

. Identify the family and the specific member.

Example 36

hard

A function in the family

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

has vertex

(-3, 5)

and passes through

(0, 14)

. Find

a

Example 37

hard

A linear and an exponential function both pass through

(0, 5)

and

(1, 10)

. Find each. At

x = 4

, which is larger?

Example 38

hard

For the family

y = a \sin(bx) + c

, which parameter changes the vertical midline?

Example 39

hard

A radioactive sample has half-life

5

years. Identify the family and write a model starting from

80

Example 40

challenge

A function in the family

y = a \cdot b^x

passes through

(2, 18)

and

(5, 486)

. Find

a

and

b

Example 41

challenge

For the family

y = a x^n

with

a > 0

and

n

a positive integer, prove every member passes through

(1, a)

and

(0, 0)

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Related Concepts

Parameter Function Parent Functions Function Transformation

Background Knowledge

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

parameterfunction definition