Lesson 20

Changes over Equal Intervals

Let's explore how linear and exponential functions change 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:

\(f(x) = x^5\)

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.

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.
Graph of 2 lines.

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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?

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:

\(q(x) = 87.5\)

(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?

(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:

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

(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. \(d(x)= \frac{x}{4}\)
(From Unit 4, Lesson 17.)