Practice Changing Rate 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 changing rate of change means the output grows by different amounts for equal increases in input — the hallmark of nonlinear functions like quadratics and exponentials.

Changing rate means accelerating or decelerating progress — like compound interest where each year's gain is larger than the last because the base keeps growing.

Showing a random 20 of 50 problems.

Example 1

easy
Outputs of a function for x=0,1,2,3x=0,1,2,3 are 1,2,4,81,2,4,8. Are first differences constant?

Example 2

medium
Compare the change of y=x2y=x^2 over [0,1][0,1] and [3,4][3,4]. Which is larger?

Example 3

easy
A car's distance is 0,10,25,450,10,25,45 km at 0,1,2,30,1,2,3 hours. Is the speed constant?

Example 4

easy
Does compound interest grow at a constant or changing rate?

Example 5

medium
Why can't you use a single slope to describe y=x2y=x^2 everywhere?

Example 6

challenge
Population doubles every 33 years from 10001000. Find the average annual rate of change over the first 33 years versus the next 33 years.

Example 7

medium
For f(x)=2xf(x)=2^x, find the average rate of change on [0,3][0,3].

Example 8

hard
A function satisfies f(0)=2f(0)=2, f(3)=14f(3)=14, f(6)=50f(6)=50. Compare the average rates on [0,3][0,3] and [3,6][3,6].

Example 9

medium
For f(x)=x2−2xf(x)=x^2-2x, find the average rate on [a,a+1][a,a+1].

Example 10

hard
A function has constant second differences of 22. Is the rate of change of ff constant?

Example 11

challenge
For y=x2y=x^2, compute average rates over [1,2][1,2], [1,1.5][1,1.5], [1,1.1][1,1.1]. What value do they approach?

Example 12

challenge
For f(x)=x3f(x)=x^3, find aa such that the average rate of change on [0,a][0,a] equals 99.

Example 13

medium
Outputs are 2,2,2,22,2,2,2 for equal input steps. Is the rate changing?

Example 14

easy
For f(x)=x2f(x)=x^2, find the average rate of change on [2,5][2,5].

Example 15

medium
Is a positive but decreasing rate the same as the output decreasing?

Example 16

easy
True or false: a linear function has a changing rate of change.

Example 17

easy
Does a curved (non-straight) graph have a constant or changing rate?

Example 18

medium
A population grows from 200200 to 450450 in 55 years. Find the average rate of change per year.

Example 19

hard
For g(x)=x3g(x) = x^3, find the average rate of change on [a,a+h][a, a+h] and simplify to see what happens as h→0h \to 0.

Example 20

medium
For f(x)=x2+1f(x)=x^2+1, compute the average rate of change on [1,4][1,4] and on [1,3][1,3]. What pattern do you notice?
← Back to Changing Rate Worked Examples Formula Explained