Practice Multiple Representations in Math

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

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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=2xy = 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 hh for 0≤h≤120\le h\le 12, and find the hour at which the cap kicks in.

Example 2

easy
Describe y=x+1y=x+1 in words.

Example 3

hard
From the table (1,1),(2,8),(3,27),(4,64)(1,1),(2,8),(3,27),(4,64), identify the function family and write the formula.

Example 4

easy
Evaluate y=2xy=2x at x=0x=0 to find where the graph crosses the yy-axis.

Example 5

medium
A line on a graph rises 33 units for every 11 unit right and crosses the yy-axis at −2-2. 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 0,1,20,1,2 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=0,1,2,3t=0,1,2,3 hours, and find the time when the population first exceeds 10001000.

Example 8

hard
A function table: (0,1),(1,3),(2,9),(3,27)(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)(2,5) with slope −12-\tfrac{1}{2}. Write its equation, find the xx-intercept, and the yy-intercept.

Example 10

medium
A function is 'subtract 2, then triple.' Write the formula and find f(5)f(5).

Example 11

medium
A graph shows a parabola opening upward with vertex at the origin. Which formula could it be: y=x2y=x^2 or y=2xy=2x?

Example 12

medium
For f(x)=x2f(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)(2,-1) and passing through (0,3)(0,3). Write the formula in vertex form.

Example 15

challenge
A function has table (0,1),(1,2),(2,4),(3,8)(0,1),(1,2),(2,4),(3,8). Decide if it is linear or exponential, give the formula, and predict yy at x=5x=5.

Example 16

medium
From the table (0,5),(1,3),(2,1),(3,−1)(0,5),(1,3),(2,1),(3,-1), write the formula and find yy at x=10x=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=0,1,2t=0,1,2, and find when distance is 4040 m.

Example 18

easy
From the table (1,3),(2,6),(3,9)(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+by=mx+b and passes through (2,7)(2,7) and (5,16)(5,16). Find mm and bb.
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Related Concepts

FunctionCoordinate Plane