Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

An exponential function is written in the form $f (x) = a b^{x}$ , where $a$ is the initial value and $b$ is the growth factor.

1. In a biology lab, a population of 50 bacteria reproduce by splitting. Every hour, on the hour, each bacterium splits into two bacteria.

a. Write a sentence of the form “_____ is a function of _____.”

Compare your answer:

The number of bacteria is a function of time passed in hours.

b. Indicate which is the independent and which is the dependent variable.

Compare your answer:

Independent: number of hours. Dependent: number of bacteria.

c. Write an equation that represents the situation using function notation.

Compare your answer:

$f (t) = 50 \cdot 2^{t}$

2. Every year after a new car is purchased, it loses $\frac{1}{3}$ of its value. Let’s say that the new car costs $18,000.

a. Write a sentence of the form “_____ is a function of _____.”

Compare your answer:

The value of a car is a function of the number of years since purchase.

b. Indicate which is the independent and which is the dependent variable.

Compare your answer:

Independent: number of years since purchase of car. Dependent: value of car in dollars.

c. Write an equation that represents the situation using function notation.

Compare your answer:

$f (t) = 18, 000 \cdot (\frac{2}{3})^{t}$ . Note: $\frac{1}{3}$ is the rate, and growth factor $= 1 -$ rate, so the growth factor is $\frac{2}{3}$ because there’s $\frac{2}{3}$ left.

3. To control an algae bloom in a lake, scientists introduce some treatment products. The day they begin treatment, the area covered by algae is 240 square yards. Each day since the treatment began, a third of the previous day’s area (in square yards) remains covered by algae. Time $t$ is measured in days.

a. Write a sentence of the form “_____ is a function of _____.”

Compare your answer:

The area covered by algae is a function of the number of days since the treatment.

b. Indicate which is the independent and which is the dependent variable.

Compare your answer:

Independent: number of days since treatment. Dependent: area of algae in square yards.

c. Write an equation that represents the situation using function notation.

Compare your answer:

$f (t) = 240 \cdot (\frac{1}{3})^{t}$ . Note: $\frac{1}{3}$ is given as the growth factor because it represents what remains.

Video: Writing an Exponential Function from a Situation

Watch the following video to learn more about exponential functions:

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Self Check

A new computer is purchased for $2,500. Each year after its purchase, it loses

\frac15

of its value. Which equation represents this situation?

$f(t)=\frac{4}{5} \cdot (2500)^t$
$f(t)=2500 \cdot (\frac{4}{5})^t$
$f(t)=2500 \cdot (\frac{4}{5})t$
$f(t)=2500+(\frac{4}{5})^t$

Additional Resources

Writing Functions from Exponential Situations

EXPONENTIAL GROWTH

A function that models exponential growth grows by a rate proportional to the amount present. For any real number $x$ and any positive real numbers $a$ and $b$ such that $b \neq 1$ , an exponential growth function has the form

$f (x) = a b^{x}$

where

$a$ is the initial or starting value of the function.
$b$ is the growth factor or growth multiplier per unit $x$ .

Example 1

The starting value of the function is 10. This function doubles every year.

Growth factor function	$𝑓 (𝑥) = 𝑎 b^{x}$
Identify the initial value.	$a = 10$
Identify the growth factor.	$b = 2$
Identify the unit.	$x = t$ for time in years.
Write the exponential equation.	$𝑓 (y) = 10 \cdot 2^{t}$

Example 2

An acre of land is worth $40,000 at the time of purchase. It doubles in value every decade. Write an equation for this situation where $t$ is time in decades.

Identify the initial value.	$a = 40, 000$
Identify the growth factor.	$b = 2$
Write the exponential equation.	$f (t) = 40, 000 \cdot 2^{t}$

Try it

Try It: Writing Functions from Exponential Situations

1. The starting value of the function is 150. This function triples every decade

Compare your answer:

Here is how to write an exponential function from a situation:

Identify the initial value.	$a = 150$
Identify the growth factor.	$b = 3$
Write the exponential equation.	$f (y) = 150 \cdot 3^{t}$

2. When first measured, the population of deer in a forest is 40. Over time, the population triples every year. Write an exponential equation where $t$ is time in years.

Compare your answer:

Here is how to write an exponential function from a situation:

Identify the initial value.	$a = 40$
Identify the growth factor.	$b = 3$
Write the exponential equation.	$f (t) = 40 \cdot 3^{t}$

5.8.3 Using Function Language and Notation

Activity

Video: Writing an Exponential Function from a Situation

Additional Resources

Writing Functions from Exponential Situations

Try It: Writing Functions from Exponential Situations