Interpreting Negative Exponents in Exponential Growth

5.6.2 Interpreting Negative Exponents in Exponential Growth

5.6.2 • Interpreting Negative Exponents in Exponential Growth

Activity

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

Two clownfish swim among the pale pink tentacles of sea anemones in a colorful coral reef aquarium, surrounded by various soft corals and marine life.

1. Write an equation of the form $y = a \cdot b^{t}$ 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.)

Compare your answer:

$y = 1, 200 \cdot 2^{t}$

2. Find the volume of the coral when $t$ is 5, 1, 0, –1, and –2. Note that when $t = - 1$ , this represents one year before the marine biologist made the original estimate.

Time (years)	5	1	0	-1	-2
Volume of Coral (cubic centimeters)	a.____	b.____	c.____	d.____	e.____

This table displays time in years and corresponding placeholder values for coral volume, spanning past, present, and future time points.

a. Enter the volume of coral when t = 5.

3800 cubic centimeters

b. Enter the volume of coral when t = 1.

2400 cubic centimeters

c. Enter the volume of coral when t = 0.

1200 cubic centimeters

d. Enter the volume of coral when t = -1.

600 cubic centimeters

e. Enter the volume of coral when t = -2.

300 cubic centimeters

3. What does it mean, in this situation, when $t$ is –2?

Compare your answer:

This represents the time 2 years before the scientist made the estimate of 1,200 cubic centimeters.

4. In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Be prepared to show your reasoning.

Compare your answer:

$t$ is –5, or 5 years before the scientist made the estimate of 1,200 cubic centimeters.

Self Check

Using the equation

p=10,000\cdot 2^d

, find the value of

p

when

d = -2

40,000
5,000
20,000
2,500

Additional Resources

Interpreting Negative Exponents in Exponential Growth

Let’s look at a problem involving an exponential growth equation and negative exponents.

A forest fire has been burning for several days. The burned area, in acres, is given by the equation $y = 4800 \cdot 2^{d}$ , where $d$ is the number of days since the area of the fire was first measured.

1. Let’s complete the table.

$d$ , days since first measurement	Process column	$y$ , acres burned since fire started
0	$= 4800 \cdot 2^{0} = 4800 \cdot 1$	4800
–1	$= 4800 \cdot 2^{1} = 4800 \cdot \frac{1}{2}$	2400
–2	0	c. _____
–3	0	d. _____
–5	0	e. _____

To determine the value of $y$ , given $d$ , follow these steps:

Example 1

Determine the value for $y$ when $d = - 2$

Step 1 -Substitute the given domain value for d into the expression.

$4800 \cdot 2^{- 2}$

Step 2 -Evaluate the exponential value. Recall that a negative exponent represents how many times the reciprocal of the base should be multiplied.

$= 4800 \cdot (\frac{1}{2})^{2} = 4800 \cdot \frac{1}{4}$

Step 3 -Evaluate the remaining expression to determine $y$ .

$= 1200$

Example 2

Determine the value for $y$ when $d = - 3$

Step 1 -Substitute the given domain value for d into the expression.

$4800 \cdot 2^{- 3}$

Step 2 -Evaluate the exponential value. Recall that a negative exponent represents how many times the reciprocal of the base should be multiplied.

$= 4800 \cdot (\frac{1}{2})^{3} = 4800 \cdot \frac{1}{8}$

Step 3 -Evaluate the remaining expression to determine $y$ .

$= 600$

Example 3

Determine the value for $y$ when $d = - 5$

Step 1 -Substitute the given domain value for d into the expression.

$4800 \cdot 2^{- 5}$

Step 2 -Evaluate the exponential value. Recall that a negative exponent represents how many times the reciprocal of the base should be multiplied.

$= 4800 \cdot (\frac{1}{2})^{5} = 4800 \cdot \frac{1}{32}$

Step 3 -Evaluate the remaining expression to determine $y$ .

$= 150$

Example 4

What does the value of $y$ tell you about the area burned in the fire when $d = - 1$ ?

The $y$ value at $d = - 1$ represents the fact that 2,400 acres had burned 1 day before the first measurement was taken.

Example 5

How much area had the fire burned a week before it measured 4,800 acres? Explain your reasoning.

Since there are 7 days in a week, to find the area burned a week before the fire was measured, we need to determine the value of $y$ when $d = - 7$ .

Step 1 -Substitute the given domain value for d into the expression.

$4800 \cdot 2^{- 7}$

Step 2 -Evaluate the exponential value. Recall that a negative exponent represents how many times the reciprocal of the base should be multiplied.

$= 4800 \cdot {\frac{1}{2}}^{7} = 4800 \cdot \frac{1}{128}$

Step 3 -Evaluate the remaining expression to determine $y$ .

$= 37.5$

$37.5$ acres had burned a week before the fire measured $4, 800$ acres. The value is found by substituting $- 7$ for d since that represents $7$ days before the first measurement of $4, 800$ acres.

Try it

Interpreting Negative Exponents in Exponential Growth

The value of a home in 2015 was $400,000. Its value has doubled each decade.

1. If $v$ is the value of the home, in dollars, write an equation for $v$ in terms of $d$ , the number of decades since 2015.

Compare your answers:

$v = 400, 000 \cdot 2^{d}$

2. What is $v$ when $d = - 1$ ?

$v = 200, 000$ $v = 400, 000 \cdot 2^{- 1}$ $v = 400, 000 \cdot 2^{- 1}$ $v = 400, 000 \cdot \frac{1}{2}$ $v = 400, 000 \cdot \frac{1}{2}$ $v = 200, 000$ $v = 200, 000$

3. What does this value mean?

Compare your answers:

This means the home had a value of 50,000 in 1985, three decades before 2015.

4. What does this value mean?

Compare your answers:

This means the home had a value of 200,000 in 2005, one decade before 2015.

5. What is $v$ when $d = - 3$ ?

$v = 50, 000$ $v = 400, 000 \cdot 2^{- 3}$ $v = 400, 000 \cdot 2^{- 3}$ $v = 400, 000 \cdot \frac{1}{8}$ $v = 400, 000 \cdot \frac{1}{8}$ $v = 50, 000$ $v = 50, 000$