# 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 \(f\)?

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

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

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

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

### 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.

When the input \(x\) increases by 1, the value of \(f\) increases by 2.

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

When the input \(x\) increases by 3, the value of \(f\) increases by 8.

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

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 \(g\).

- Use the graph to explain why the value of \(f\) increases by 2 each time the input \(x\) increases by 1.
- 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 \(h(x) = 5^x\).

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

\(f(x) = x + 2\)

\(g(x) = 2^x\)

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

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

\(q(x) = 87.5\)

### 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?

### 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?

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

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

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

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

### 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.

- \(a(x)=x-4\)
- \(b(x)=2x-4\)
- \(c(x)=2(x-4)\)
- \(d(x)= \frac{x}{4}\)