# Lesson 19

Which One Changes Faster?

Let's compare linear and exponential functions as they continue to increase.

### Problem 1

Functions $$a,b,c,d,e,$$ and $$f$$ are given below. Classify each function as linear, exponential, or neither.

1. $$a(x)=3x$$
2. $$b(x)=3^x$$
3. $$c(x)=x^3$$
4. $$d(x)=9+3x$$
5. $$e(x)=9\boldcdot3^x$$
6. $$f(x)=9\boldcdot3x$$

### Problem 2

Here are 4 equations defining 4 different functions, $$a, b, c,$$ and $$d$$. List them in order of increasing rate of change. That is, start with the one that grows the slowest and end with the one that grows the quickest.

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

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

$$c(x)=x+4$$

$$d(x)=1+4x$$

### Problem 3

Technology required. Function $$f$$ is defined by $$f(x) = 3x + 5$$ and function $$g$$ is defined by $$g(x) = (1.1)^x$$.

1. Complete the table with values of $$f(x)$$ and $$g(x)$$. When necessary, round to 2 decimal places.
2. Which function do you think grows faster? Explain your reasoning.
3. Use technology to create graphs representing $$f$$ and $$g$$. What graphing window do you have to use to see the value of $$x$$ where $$g$$ becomes greater than $$f$$ for that $$x$$?
$$x$$   $$f(x)$$     $$g(x)$$
1
5
10
20

### Problem 4

Functions $$m$$ and $$n$$ are given by $$m(x)=(1.05)^x$$ and $$n(x)=\frac58 x$$. As $$x$$ increases from 0:

1. Which function reaches 30 first?
2. Which function reaches 100 first?

### Problem 5

The functions $$f$$ and $$g$$ are defined by $$f(x) = 8x + 33$$ and $$g(x) = 2 \boldcdot (1.2)^x$$.

1. Which function eventually grows faster, $$f$$ or $$g$$? Explain how you know.
2. Explain why the graphs of $$f$$ and $$g$$ meet for a positive value of $$x$$.

### Problem 6

A line segment of length $$\ell$$ is scaled by a factor of 1.5 to produce a segment with length $$m$$. The new segment is then scaled by a factor of 1.5 to give a segment of length $$n$$.

What scale factor takes the segment of length $$\ell$$ to the segment of length $$n$$? Explain your reasoning.

(From Unit 5, Lesson 16.)

### Problem 7

A couple needs to get a loan of \$5,000 and has to choose between three options.

• Option A: $$2\frac{1}{4}\%$$ applied quarterly
• Option B: $$3\%$$ applied every 4 months
• Option C: $$4\frac{1}{2}\%$$ applied semi-annually

If they make no payments for 5 years, which option will give them the least amount owed after 5 years? Use a mathematical model for each option to explain your choice.

(From Unit 5, Lesson 17.)

### Problem 8

Here are graphs of five absolute value functions. Match the graph and equation that represent the same function.

(From Unit 4, Lesson 14.)