Lesson 7

Using Negative Exponents

Let’s look more closely at exponential graphs and equations.

7.1: Exponent Rules

How would you rewrite each of the following as an equivalent expression with a single exponent?

  • \(2^4 \boldcdot 2^0\)
  • \(2^4 \boldcdot 2^{\text-1}\)
  • \(2^4 \boldcdot 2^{\text-3}\)
  • \(2^4 \boldcdot 2^{\text-4}\)

7.2: Coral in the Sea

A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.

Coral underwater with fish.
  1. Write an equation of the form \(y=a \boldcdot b^t\) representing the relationship, where \(t\) is time in years since the coral was measured and \(y\) is volume of coral in cubic centimeters. (You need to figure out what numbers \(a\) and \(b\) are in this situation.)
  2. Find the volume of the coral when \(t\) is 5, 1, 0, -1, and -2.
  3. What does it mean, in this situation, when \(t\) is -2?
  4. In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Explain your reasoning.

7.3: Windows of Graphs

The volume, \(y\), of coral in cubic centimeters is modeled by the equation \(y = 1,\!200 \boldcdot 2^x\) where \(x\) is the number of years since the coral was measured. Three students used graphing technology to graph the equation that represents the volume of coral as a function of time.

A

Two points on coordinate plane, origin O.

B

Graph of function on a grid.

C

graph of function on grid.

For each graph:

  1. Describe how well each graphing window does, or does not, show the behavior of the function.
  2. For each graphing window you think does not show the behavior of the function well, describe how you would change it.
  3. Make the change(s) you suggested, and sketch the revised graph using graphing technology.

7.4: Measuring Meds

A person took some medicine but does not remember how much. Concerned that she took too much, she has a blood test every hour for several hours.

    1. Time \(t\) is measured in hours since the blood test and amount of medicine in her body, \(m\), is measured in milligrams. What is the growth factor? That is, what is \(b\) in an equation of the form \(m=a \boldcdot b^t\) ? What is \(a\)?
    2. Find the amounts of medicine in the patient’s body when \(t\) is -1 and -3. Record them in the table.
    \(t\), time (hours) \(m\), medicine (mg)
       
       
    0 100
    1 50
    2 25
  1. What do \(t=0\) and \(t=\text-3\) mean in this context?
  2. The medicine was taken when \(t\) is -5. Assuming the person did not have any of the medication in her body beforehand, how much medicine did the patient take?
  3. Plot the points whose coordinates are shown in the table. Make sure to draw and label tick marks on the axes.
    Blank coordinate plane, no grid, Origin O. Horizontal axis labeled “t”. Vertical axis, labeled “m”. Vertical axis intersects horizontal axis at origin.
 
  4. Based on your graph, when do you think the patient will have:

    1. 500 mg of medicine remaining in the body
    2. no medicine remaining in the body


Without evaluating them, describe each of the following quantities as close to 0, close to 1, or much larger than 1.

\(\displaystyle \frac{1}{1-2^{\text-10}}\qquad\qquad \frac{2^{10}}{2^{10}+1}\qquad\qquad \frac{2^{\text-10}}{2^{10}+1}\qquad\qquad \frac{1-2^{\text-10}}{2^{10}} \qquad\qquad \frac{1+2^{10}}{2^{\text-10}}\)

 

Summary

Equations are useful not only for representing relationships that change exponentially, but also for answering questions about these situations.

Suppose a bacteria population of 1,000,000 has been increasing by a factor of 2 every hour. What was the size of the population 5 hours ago? How many hours ago was the population less than 1,000?

We could go backwards and calculate the population of bacteria 1 hour ago, 2 hours ago, and so on. For example, if the population doubled each hour and was 1,000,000 when first observed, an hour before then it must have been 500,000, and two hours before then it must have been 250,000, and so on.

Another way to reason through these questions is by representing the situation with an equation. If \(t\) measures time in hours since the population was 1,000,000, then the bacteria population can be described by the equation:

\(\displaystyle p = 1,\!000,\!000 \boldcdot 2^{\text-t}\)

The population is 1,000,000 when \(t\) is 0, so 5 hours earlier, \(t\) would be -5 and here is a way to calculate the population:

\(\displaystyle \begin{align} 1,\!000,\!000 \boldcdot 2^{\text-5} &= 1,\!000,\!000 \boldcdot \frac{1}{2^5} \\ &= 1,\!000,\!000 \boldcdot \frac{1}{32} \\ &= 31,\!250 \end{align}\)

Likewise, substituting -10 for \(t\) gives us \(1,\!000,\!000 \boldcdot 2^{\text-10}\) or \(1,\!000,\!000 \boldcdot \frac{1}{2^{10}}\), which is a little less than 1,000. This means that 10 hours before the initial measurement the bacteria population was less than 1,000.

Glossary Entries

  • growth factor

    In an exponential function, the output is multiplied by the same factor every time the input increases by one. The multiplier is called the growth factor.