Changes over Equal Intervals

ID: variz-jikon
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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19 questions

Changes over Equal Intervals

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Problem 1

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

a)

ff(xx)=3xx+5

b)

ff(xx)=5xx+3

c)

ff(xx)=5x5^{x}

d)

ff(xx)=x5x^{5}

Problem 2

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

a) When the input xx increases by 1, the value of ff increases by 2.

b) When the input xx increases by 1, the value of ff increases by a factor of 2.

c) When the input xx increases by 3, the value of ff increases by 8.

d) When the input xx increases by 3, the value of ff increases by a factor of 8.

e) When the input xx increases by 4, the value of ff increases by a factor of 4.

Problem 3

The two lines on the coordinate plane are graphs of functions ff and gg.

A template for answering this question. Ask your instructor for an alternative.

3) Use the graph to explain why the value of ff increases by 2 each time the input xx increases by 1.

4) Use the graph to explain why the value of gg increases by 2 each time the input xx increases by 1.

Problem 4

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

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

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

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

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

Problem 5

9) For each of the functions f,g,h,p,f, g, h, p, and qq, the domain is 0x1000 \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 all\textbf{all} that apply. Write each corresponding letter in the answer box and separate letters with commas.

a) f(x)=x+2f(x) = x + 2 \quad\quad b) g(x)=2xg(x) = 2^x \quad\quad c) h(x)=111x23h(x) = 111x - 23 \quad\quad d) p(x)=50,0003xp(x) = 50,000 \cdot 3^x \quad\quad e) q(x)=87.5q(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?

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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)^{3}

b)

500(1+0.144)3\cdot\left(1+\frac{0.14}{4}\right)^{3}

c)

500(1+0.144)12\cdot\left(1+\frac{0.14}{4}\right)^{12}

d)

500(1+0.144)48\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)=x4a(x) = x - 4

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

13) Write the inverse function.

b(x)=2x4b(x) = 2x - 4

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

15) Write the inverse function.

c(x)=2(x4)c(x) = 2(x - 4)

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

17) Write the inverse function.

d(x)=x4d(x) = \frac{x}{4}

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

19) Write the inverse function.