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Algebra 1

5.6.2 Interpreting Negative Exponents in Exponential Growth

Algebra 15.6.2 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.

1. Write an equation of the form y = a · b t y = a · b t representing the relationship, where t t  is time in years since the coral was measured and y y  is volume of coral in cubic centimeters. (You need to figure out what numbers a a and b b  are in this situation.)

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

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

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

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

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

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

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

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.

Self Check

Using the equation p = 10 , 000 ⋅ 2 d , find the value of p when d = − 2 .
  1. 40,000
  2. 5,000
  3. 20,000
  4. 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 · 2 d y = 4800 · 2 d , where d d is the number of days since the area of the fire was first measured.

1. Let’s complete the table.

d d , days since first measurement Process column y y , acres burned since fire started
0 = 4800 · 2 0 = 4800 · 1 = 4800 · 2 0 = 4800 · 1 4800
–1 = 4800 · 2 1 = 4800 · 1 2 = 4800 · 2 1 = 4800 · 1 2 2400
–2 0 c. _____
–3 0 d. _____
–5 0 e. _____

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

Example 1

Determine the value for y y when d = − 2 d = − 2

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

4800 · 2 − 2 4800 · 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 · ( 1 2 ) 2 = 4800 · 1 4 = 4800 · ( 1 2 ) 2 = 4800 · 1 4

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

= 1200 = 1200

Example 2

Determine the value for y y when d = − 3 d = − 3

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

4800 · 2 − 3 4800 · 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 · ( 1 2 ) 3 = 4800 · 1 8 = 4800 · ( 1 2 ) 3 = 4800 · 1 8

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

= 600 = 600

Example 3

Determine the value for y y when d = − 5 d = − 5

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

4800 · 2 − 5 4800 · 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 · ( 1 2 ) 5 = 4800 · 1 32 = 4800 · ( 1 2 ) 5 = 4800 · 1 32

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

= 150 = 150

Example 4

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

The y y value at d = − 1 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 y when d = − 7 d = − 7 .

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

4800 · 2 − 7 4800 · 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 · 1 2 7 = 4800 · 1 128 = 4800 · 1 2 7 = 4800 · 1 128

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

= 37.5 = 37.5

37.5 37.5 acres had burned a week before the fire measured 4 , 800 4 , 800 acres. The value is found by substituting − 7 − 7 for d since that represents 7 7 days before the first measurement of 4 , 800 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 v is the value of the home, in dollars, write an equation for v v in terms of d d , the number of decades since 2015.

2. What is v v when d = − 1 d = − 1 ?

3. What does this value mean?

4. What does this value mean?

5. What is v v when d = − 3 d = − 3 ?

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