# Lesson 12

Reasoning about Exponential Graphs (Part 1)

Let’s study and compare equations and graphs of exponential functions.

Jada received a gift of \$180. In the first week, she spent a third of the gift money. She continues spending a third of what is left each week thereafter. Which equation best represents the amount of gift money $$g$$, in dollars, she has after $$t$$ weeks? Be prepared to explain your reasoning.

1. $$g = 180 - \frac13 t$$
2. $$g = 180 \boldcdot \left(\frac13\right)^t$$
3. $$g = \frac13 \boldcdot 180^t$$
4. $$g = 180 \boldcdot \left(\frac23\right)^t$$

### 12.2: Equations and Their Graphs

1. Each of the following functions $$f$$, $$g$$$$,$$ $$h$$, and $$j$$ represents the amount of money in a bank account, in dollars, as a function of time $$x$$, in years. They are each written in form $$m(x) = a \boldcdot b^x$$.
$$\displaystyle f(x) = 50 \boldcdot 2^x$$
$$\displaystyle g(x) = 50 \boldcdot 3^x$$
$$\displaystyle h(x) = 50 \boldcdot \left(\frac32 \right)^x$$
$$\displaystyle j(x) = 50 \boldcdot (0.5)^x$$

1. Use graphing technology to graph each function on the same coordinate plane.
2. Explain how changing the value of $$b$$ changes the graph.
2. Here are equations defining functions $$p$$, $$q$$, and $$r$$. They are also written in the form $$m(x) = a \boldcdot b^x$$.
$$\displaystyle p(x) = 10 \boldcdot 4^x$$
$$\displaystyle q(x) = 40 \boldcdot 4^x$$
$$\displaystyle r(x) = 100 \boldcdot 4^x$$

1. Use graphing technology to graph each function and check your prediction.
2. Explain how changing the value of $$a$$ changes the graph.

As before, consider bank accounts whose balances are given by the following functions:

$$\displaystyle f(x)=10\boldcdot 3^x \qquad\qquad g(x)=3^{x+2}\qquad\qquad h(x)=\tfrac{1}{2}\boldcdot 3^{x+3}$$

Which function would you choose? Does your choice depend on $$x$$?

### 12.3: Graphs Representing Exponential Decay

$$\displaystyle m(x) = 200 \boldcdot \left(\frac14 \right)^x$$
$$\displaystyle n(x) = 200 \boldcdot \left(\frac12 \right)^x$$
$$\displaystyle p(x) = 200 \boldcdot \left(\frac34 \right)^x$$
$$\displaystyle q(x) = 200 \boldcdot \left(\frac78 \right)^x$$

1. Match each equation with a graph. Be prepared to explain your reasoning.

2. Functions $$f$$ and $$g$$ are defined by these two equations: $$f(x) = 1,\!000 \boldcdot \left( \frac{1}{10} \right)^x$$ and $$g(x) = 1,\!000 \boldcdot \left( \frac{9}{10} \right)^x$$.

1. Which function is decaying more quickly? Explain your reasoning.
2. Use graphing technology to verify your response.

### Summary

An exponential function can give us information about a graph that represents it.

For example, suppose the function $$q$$ represents a bacteria population $$t$$ hours after it is first measured and $$q(t) = 5,\!000 \boldcdot (1.5)^t$$. The number 5,000 is the bacteria population measured, when $$t$$ is 0. The number 1.5 indicates that the bacteria population increases by a factor of 1.5 each hour.

A graph can help us see how the starting population (5,000) and growth factor (1.5) influence the population. Suppose functions $$p$$ and $$r$$ represent two other bacteria populations and are given by $$p(t) = 5,\!000 \boldcdot 2^t$$ and  $$r(t) =5,\!000 \boldcdot (1.2)^t$$. Here are the graphs of $$p$$, $$q$$, and $$r$$.

All three graphs start at $$5,\!000$$ but the graph of $$r$$ grows more slowly than the graph of $$q$$ while the graph of $$p$$ grows more quickly. This makes sense because a population that doubles every hour is growing more quickly than one that increases by a factor of 1.5 each hour, and both grow more quickly than a population that increases by a factor of 1.2 each hour.
An exponential function is a function that has a constant growth factor. Another way to say this is that it grows by equal factors over equal intervals. For example, $$f(x)=2 \boldcdot 3^x$$ defines an exponential function. Any time $$x$$ increases by 1, $$f(x)$$ increases by a factor of 3.