Exponential Change: The Growth Factor

5.4.3 Exponential Change: The Growth Factor

5.4.3 • Exponential Change: The Growth Factor

Activity

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.

1. In a biology lab, 500 bacteria reproduce by splitting. Every hour, on the hour, each bacterium splits into two bacteria. Fill in the table by writing an expression to show the number of bacteria after each hour.

hour	number of bacteria
0	500
1	a. _____
2	b. _____
3	c. _____
6	d. _____
$t$	e. _____

a. _____

$500 + 2$

$500 \cdot 2$

b. _____

$500 + 2 + 2$

$500 \cdot 2 \cdot 2$

c. _____

$500 + 2 + 2 + 2$

$500 \cdot 2 \cdot 2 \cdot 2$

The answer could also be written $500 \cdot 2^{3}$ .

d. _____

$500 + 2 + 2 + 2 + 2$

$500 + 2 + 2 + 2 + 2 + 2 + 2$

$500 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

$500 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

The answer could also be written $500 \cdot 2^{6}$ .

e. _____

$500 + 2 t$

$500 + 2^{t}$

$500 \cdot 2^{t}$

$500 \cdot 2 \cdot t$

$500 \cdot 2^{t}$

2. Which equation relates $n$ , the number of bacteria, to $t$ , the number of hours?

$n = 500 + 2 t$
$n = 500 + 2^{t}$
$n = 500 \cdot 2^{t}$
$n = 500 \cdot 2 \cdot t$

$n = 500 \cdot 2^{t}$

3. Use your equation to find $n$ when $t$ is 0. What does this value of $n$ mean in this situation?

Compare your answer:

500; When $t$ is 0, $n = 500 \cdot 2^{0}$ . Because $2^{0} = 1$ , $n = 500 \cdot 1$ or $n = 500$ . 500 is the number of bacteria at the starting time or at hour 0.

4. In a different biology lab, a population of single-cell parasites also reproduces hourly. An equation that gives the number of parasites, $p$ , after $t$ hours is $p = 100 \cdot 3^{t}$ . Explain what the numbers 100 and 3 mean in this situation.

Compare your answer:

100 is the number of parasites at hour 0 because $100 \cdot 3^{0} = 100$ . The number 3 means that every hour, the number of parasites triples.

Self Check

The population of a city is 100,000. It doubles each decade. Write an equation

p

that relates the population to

t

, time in decades.

$p=2^t$
$p=100,000\cdot t^2$
$p=100,000\cdot2^t$
$p=100,000\cdot2t$

Additional Resources

Identifying the Constant Ratio of the Exponential Function

What is the constant ratio (also called the growth factor) of the exponential function shown in the table? Rewrite each term to show the initial value and repeated use of the constant ratio.

$x$	0	1	2	3	4
$f (x)$	4	12	36	108	324

This table displays values of variable x and corresponding values of the function f(x) for specific inputs.

Defining Exponential Growth

Because the output of exponential functions increases very rapidly, the term “exponential growth” is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.

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 $x$ .

The initial or starting value of the function is 4, because that is when the value of $x$ is 0.

The constant ratio, $b$ , would be 3, since every value in the table is 3 times the previous term.

$x$	0	1	2	3	4
$f (x)$	4	$4 \cdot 3^{1}$	$4 \cdot 3^{2}$	$4 \cdot 3^{3}$	$4 \cdot 3^{4}$

Values for x and f(x) for the exponential function f(x) = 4 * 3^x from x=0 to x=4.

Try it

Identifying the Constant Ratio of the Exponential Function

What is the constant ratio of the following exponential function?

$x$	1	2	3	4
$f (x)$	10	20	40	80

This table presents input values 'x' and their corresponding output values 'f(x)', demonstrating an exponential function's behavior.

Compare your answer:

Here is how to identify the exponential constant ratio of a table:

First, make sure the table is reflecting an exponential relationship by noticing how quickly the terms grow. Then, if the $x$ values are increasing by 1, divide consecutive terms to determine the growth factor.

Since $20 \div 10 = 2$ , $40 \div 20 = 2$ , and $80 \div 40 = 2$ , the growth factor is 2.