Multiple Representations Examples: 46 Problems with Answers

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

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.'

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: Every function can appear as a formula, a table, a graph, or words — all describing the identical rule.

Common stuck point: The procedure for multiple representations is the easy part; the trap is treating equivalent forms as different functions. Asking "Do these different-looking forms encode the exact same input-output pairs?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do these different-looking forms encode the exact same input-output pairs?

Worked Examples

Example 1

easy

Represent the function

f(x) = 2x + 1

in four ways: equation, table of values, graph description, and verbal description.

Answer

All four representations describe the same linear function

f(x)=2x+1

First step

Equation:

f(x) = 2x + 1

Full solution

2
Table: $x=-1 \to -1$ ; $x=0 \to 1$ ; $x=1 \to 3$ ; $x=2 \to 5$ .
3
Graph: a straight line with slope $2$ and $y$ -intercept $(0,1)$ , rising steeply left to right.
4
Verbal: 'Start with any number, multiply it by two, then add one.'

Multiple representations highlight different aspects of the same function. Tables reveal specific values; graphs show shape and trends; equations allow computation; verbal descriptions communicate meaning. Fluency across all four is essential in mathematics.

Example 2

medium

A table gives values:

x: 0,1,2,3

and

f(x): 1,2,4,8

. Identify the function type, write its equation, and describe its graph.

Example 3

easy

Convert the formula

f(x)=4x

into a table for

x=-1,0,1,2

and describe the graph's slope.

Example 4

medium

Convert the verbal rule 'the output is the input squared minus the input' into a formula, then build the table for

x=-1,0,1,2,3

Example 5

hard

A graph shows a line through

(-3,7)

and

(2,-3)

. Write the formula, give the

y

-intercept, and convert to a verbal rule.

Example 6

hard

A function table:

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

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

Practice Problems

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

Example 1

easy

Convert the verbal description to an equation: 'The output is three less than twice the square of the input.' Then evaluate the function at

x = 4

Example 2

hard

A graph passes through

(-2,0)

(0,-4)

, and

(1,0)

and appears to be a parabola. Find the quadratic equation and verify using all three points.

Example 3

easy

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

Example 4

easy

For

y=x+3

, fill the table value at

x=4

Example 5

easy

A line passes through

(0,0)

and

(1,2)

. What is its formula?

Example 6

easy

Describe

y=x+1

in words.

Example 7

easy

From the table

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

, what formula fits?

Example 8

easy

Plot the point for the pair

(2,5)

. What are its coordinates?

Example 9

easy

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

Example 10

easy

Evaluate

y=2x

x=0

to find where the graph crosses the

y

-axis.

Example 11

medium

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

f(5)

Example 12

medium

A table shows

(0,2),(1,5),(2,8)

. Write the formula and predict

y

x=4

Example 13

medium

A graph is a line through

(0,1)

and

(2,5)

. Give its formula.

Example 14

medium

Convert

f(x)=\frac{x}{2}

into a table for

x=0,2,4

and state the verbal rule.

Example 15

medium

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

y=x^2

y=2x

Example 16

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 17

medium

Given the table

(1,1),(2,4),(3,9)

, identify the formula.

Example 18

medium

A verbal rule says 'output is always 4.' State its graph type and give the formula.

Example 19

medium

A graph crosses the

y

-axis at

3

and falls

2

units per unit right. Formula?

Example 20

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 21

challenge

A line's graph fits

y=mx+b

and passes through

(2,7)

and

(5,16)

. Find

m

and

b

Example 22

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 23

easy

A function is described in words as 'add 5 to the input, then multiply by 2.' Write its formula.

Example 24

easy

A graph shows a horizontal line through

y=-2

. Write the formula.

Example 25

easy

Translate the formula

y=x-7

into a verbal description.

Example 26

medium

From the table

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

, write the formula and find

y

x=10

Example 27

medium

A graph passes through

(0,-3)

and

(4,5)

. Write the linear formula and the verbal rule.

Example 28

medium

A car rental costs $30 per day plus a $15 flat fee. Write the cost formula, build a table for

1,2,3

days, and find the cost for

10

days.

Example 29

medium

A table shows

(1,2),(2,4),(3,8),(4,16)

. State whether it is linear, exponential, or quadratic, and give the formula.

Example 30

medium

A graph is a V-shape with vertex at the origin opening upward, passing through

(1,1)

. Write the formula.

Example 31

medium

For

y=\frac{12}{x}

, complete the table at

x=1,2,3,4,6

and describe what happens to

y

x

increases.

Example 32

medium

A swimming pool fills at

50

gallons per minute starting from empty. Write the formula, give the volume at

t=8

minutes, and describe what the graph looks like.

Example 33

hard

A quadratic function has table values

(0,3),(1,4),(2,7),(3,12)

. Find the formula.

Example 34

hard

A graph shows a parabola with vertex at

(2,-1)

and passing through

(0,3)

. Write the formula in vertex form.

Example 35

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 36

hard

From the table

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

, identify the function family and write the formula.

Example 37

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 38

hard

A taxi charges $3 to start plus $1.50 per mile. Write the formula, build a table for

m=0,1,2,5,10

miles, and find how many miles a $15 fare allows.

Example 39

challenge

A graph shows a parabola through

(0,-3),(1,0),(3,0)

. Write the formula in standard form

ax^2+bx+c

and find its vertex.

Example 40

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.

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Related Concepts

Function Coordinate Plane

Background Knowledge

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

function definitioncoordinate plane