Lesson 12

Reasoning about Exponential Graphs (Part 1)

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

12.1: Spending Gift Money

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

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

Graph of four exponential function on x y grid. A. B. C. D.
  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 \(r\).

Graph of 3 functions on a grid. P, q, 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.

Glossary Entries

  • exponential function

    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.