Lesson 20

Changes over Equal Intervals

Problem 1

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

A:

$$f(x) = 3x + 5$$

B:

$$f(x) = 5x + 3$$

C:

$$f(x) = 5^x$$

D:

$$f(x) = x^5$$

Solution

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Problem 2

The function $$f$$ is defined by $$f(x) = 2^x$$. Which of the following statements is true about the values of $$f$$? Select all that apply.

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.

Solution

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Problem 3

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

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

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Solution

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Problem 4

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

1. Find the quotient $$\frac{h(x+2)}{h(x)}$$.
2. What does this tell you about how the value of $$h$$ changes when the input is increased by 2?
3. Find the quotient $$\frac{h(x+3)}{h(x)}$$.
4. What does this tell you about how the value of $$h$$ changes when the input is increased by 3?

Solution

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Problem 5

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 all that apply.

A:

$$f(x) = x + 2$$

B:

$$g(x) = 2^x$$

C:

$$h(x) = 111x - 23$$

D:

$$p(x) = 50,\!000 \boldcdot 3^{x}$$

E:

$$q(x) = 87.5$$

Solution

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(From Unit 5, Lesson 10.)

Problem 6

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?

Solution

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(From Unit 5, Lesson 16.)

Problem 7

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:

$$500\boldcdot\left(1 + 0.14\right)^3$$

B:

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

C:

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

D:

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

Solution

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(From Unit 5, Lesson 17.)

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.

1. $$a(x)=x-4$$
2. $$b(x)=2x-4$$
3. $$c(x)=2(x-4)$$
4. $$d(x)= \frac{x}{4}$$

Solution

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(From Unit 4, Lesson 17.)