Lesson 12

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. \(g = 180 \boldcdot \left(\frac23\right)^t\)

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.

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.

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.