Practice Multiple Representations 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

Every function can be expressed in four equivalent ways: as an algebraic formula, a table of input-output pairs, a graph on the coordinate plane, or a verbal description. Each representation highlights different properties and is useful in different contexts.

Same function, different views: $y = 2x$ as formula, as table, as line, as 'doubling.'

Showing a random 20 of 50 problems.

Example 1

challenge

A boat costs $200 to rent for the day plus $25 per hour, but is capped at $400. Write the cost as a piecewise function in hours

h

for

0\le h\le 12

, and find the hour at which the cap kicks in.

Example 2

easy

Describe

y=x+1

in words.

Example 3

hard

From the table

(1,1),(2,8),(3,27),(4,64)

, identify the function family and write the formula.

Example 4

easy

Evaluate

y=2x

x=0

to find where the graph crosses the

y

-axis.

Example 5

medium

A line on a graph rises

3

units for every

1

unit right and crosses the

y

-axis at

-2

. Write its formula.

Example 6

medium

From the description 'cost is 5 dollars plus 2 dollars per item,' write the formula and a table for

0,1,2

items.

Example 7

hard

A scenario reads: 'a population of bacteria doubles every hour, starting at 100.' Write the formula, build a table for

t=0,1,2,3

hours, and find the time when the population first exceeds

1000

Example 8

hard

A function table:

(0,1),(1,3),(2,9),(3,27)

. Classify it, write the formula, and describe the shape of the graph.

Example 9

hard

A line graph passes through

(2,5)

with slope

-\tfrac{1}{2}

. Write its equation, find the

x

-intercept, and the

y

-intercept.

Example 10

medium

A function is 'subtract 2, then triple.' Write the formula and find

f(5)

Example 11

medium

A graph shows a parabola opening upward with vertex at the origin. Which formula could it be:

y=x^2

y=2x

Example 12

medium

For

f(x)=x^2

the graph has what symmetry?

Example 13

easy

The rule 'double the input' is written as a formula. Write it.

Example 14

hard

A graph shows a parabola with vertex at

(2,-1)

and passing through

(0,3)

. Write the formula in vertex form.

Example 15

challenge

A function has table

(0,1),(1,2),(2,4),(3,8)

. Decide if it is linear or exponential, give the formula, and predict

y

x=5

Example 16

medium

From the table

(0,5),(1,3),(2,1),(3,-1)

, write the formula and find

y

x=10

Example 17

challenge

A scenario: 'distance starts at 10 m and increases by 5 m each second.' Express as a formula, a table for

t=0,1,2

, and find when distance is

40

Example 18

easy

From the table

(1,3),(2,6),(3,9)

, what formula fits?

Example 19

easy

Which representation best shows whether a function is increasing: formula or graph?

Example 20

challenge

A line's graph fits

y=mx+b

and passes through

(2,7)

and

(5,16)

. Find

m

and

b

← Back to Multiple Representations Worked Examples

Related Concepts

Function Coordinate Plane