# Changes over Equal Intervals

ID: variz-jikon
Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2

19 questions

# Changes over Equal Intervals

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##### Problem 1

1) Whenever the input of a function ﻿$f$﻿ increases by 1, the output increases by 5. Which of these equations could define ﻿$f$﻿?

a)

﻿$f$﻿(﻿$x$﻿)=3﻿$x$﻿+5

b)

﻿$f$﻿(﻿$x$﻿)=5﻿$x$﻿+3

c)

﻿$f$﻿(﻿$x$﻿)=﻿$5^{x}$﻿

d)

﻿$f$﻿(﻿$x$﻿)=﻿$x^{5}$﻿

##### Problem 2

2) The function ﻿$f$﻿ is defined by ﻿$f(x) = 2^x$﻿. Which of the following statements is true about the values of ﻿$f$﻿? Select ﻿$\textbf{all}$﻿ that apply. Write each corresponding letter in the answer box and separate letters with commas.

a) When the input ﻿$x$﻿ increases by 1, the value of ﻿$f$﻿ increases by 2.

b) When the input ﻿$x$﻿ increases by 1, the value of ﻿$f$﻿ increases by a factor of 2.

c) When the input ﻿$x$﻿ increases by 3, the value of ﻿$f$﻿ increases by 8.

d) When the input ﻿$x$﻿ increases by 3, the value of ﻿$f$﻿ increases by a factor of 8.

e) When the input ﻿$x$﻿ increases by 4, the value of ﻿$f$﻿ increases by a factor of 4.

##### Problem 3

The two lines on the coordinate plane are graphs of functions ﻿$f$﻿ and ﻿$g$﻿.

3) Use the graph to explain why the value of ﻿$f$﻿ increases by 2 each time the input ﻿$x$﻿ increases by 1.

4) Use the graph to explain why the value of ﻿$g$﻿ increases by 2 each time the input ﻿$x$﻿ increases by 1.

##### Problem 4

The function ﻿$h$﻿ is given by ﻿$h(x) = 5^x$﻿

5) Find the quotient ﻿$\frac{h(x + 2)}{h(x)}$﻿.

Show Work

6) What does this tell you about how the value of ﻿$h$﻿ changes when the input is increased by 2?

7) Find the quotient ﻿$\frac{h(x + 3)}{h(x)}$﻿.

Show Work

8) What does this tell you about how the value of ﻿$h$﻿ changes when the input is increased by 3?

##### Problem 5

9) For each of the functions ﻿$f, g, h, p,$﻿ and ﻿$q$﻿, the domain is ﻿$0 \leq x \leq 100$﻿. For which functions is the average rate of change a good measure of how the function changes for this domain? Select ﻿$\textbf{all}$﻿ that apply. Write each corresponding letter in the answer box and separate letters with commas.

a) ﻿$f(x) = x + 2 \quad\quad$﻿ b) ﻿$g(x) = 2^x \quad\quad$﻿ c) ﻿$h(x) = 111x - 23 \quad\quad$﻿ d) ﻿$p(x) = 50,000 \cdot 3^x \quad\quad$﻿ e) ﻿$q(x) = 87.5 \quad\quad$﻿

##### Problem 6

10) The average price of a gallon of regular gasoline in 2016 was $2.14. In 2017, the average price was$2.42 a gallon—an increase of 13%.

At that rate, what will the average price of gasoline be in 2020?

Show Work
##### Problem 7

11) A credit card charges a 14% annual nominal interest rate and has a balance of \$500.

If no payments are made and interest is compounded quarterly, which expression could be used to calculate the account balance, in dollars, in 3 years?

a)

500﻿$\cdot$﻿(1+0.14﻿$)^{3}$﻿

b)

500﻿$\cdot\left(1+\frac{0.14}{4}\right)^{3}$﻿

c)

500﻿$\cdot\left(1+\frac{0.14}{4}\right)^{12}$﻿

d)

500﻿$\cdot\left(1+\frac{0.14}{4}\right)^{48}$﻿

##### Problem 8

Here are equations that define four linear functions. For each function, write a verbal description of what is done to the input to get the output, and then write the inverse function.

﻿$a(x) = x - 4$﻿

12) Write a verbal description of what is done to the input to get the output for ﻿$a(x)$﻿.

13) Write the inverse function.

﻿$b(x) = 2x - 4$﻿

14) Write a verbal description of what is done to the input to get the output for ﻿$b(x)$﻿.

15) Write the inverse function.

﻿$c(x) = 2(x - 4)$﻿

16) Write a verbal description of what is done to the input to get the output for ﻿$c(x)$﻿.

17) Write the inverse function.

﻿$d(x) = \frac{x}{4}$﻿

18) Write a verbal description of what is done to the input to get the output for ﻿$d(x)$﻿.

19) Write the inverse function.