Remember that an exponential function is any function that can be written in the form

y=abx , where

a

and

b

are constants

a \neq 0, b > 0,

and

b≠1. The variable

a

represents the initial amount (y-intercept) and

b

represents the rate of change.

Here we are going to explore how to create multiple representations of exponential functions when we are given a rate of change and y-intercept.

Two girls in a small town once shared a secret, just between the two of them. They couldn’t stand it though, and each of them told three friends. Of course, their friends couldn’t keep secrets, either, and each of them told three of their friends. Those friends told three friends, and those friends told three friends, and so on . . . and pretty soon the whole town knew the secret. There was nobody else to tell!

These girls experienced the startling effects of an exponential function. If you start with the two girls who each told three friends, you can see that they told six people or 2×3. Those six people each told three others, so that 6×3 or 2 x 3 x 3. They told 18 people. Those 18 people each told 3, so that now is 18×3 or 2 x 3 x 3 x 3 or 54 people.

In this situation, the initial value is 2 and the rate of change is 3.

You can see how this is growing and you could show the number of people told in each round of gossip with a function: y=abx where y is the number of people told, $a$ is the two girls who started the gossip, $b$ is the number of friends that they each told, and x is the number of rounds of gossip that occurred.

You could make a table of values and calculate the number of people told after each round of gossip. Use the function y=2×3x where y is the number of people told and x is the number of rounds of gossip that occurred.

x rounds of gossip	0	1	2	3	4	5
people told	2	6	18	54	162	486

Next, graph the relationship between the rounds of gossip and the number of people told.

y=2(3)^x.PNG

How can you tell if a function is an exponential function?

The pattern of change in an exponential function consists of a rate of change being continually multiplied. Exponential functions can be written in the form y=abx, where $a$ and $b$ are constants, $a \neq 0, b > 1$ , and b≠1. In exponential functions, the exponent is a variable. Their graphs will have a characteristic curve either upward or downward and their table will show a pattern of repeated multiplication.

Examples

Example 1

In a laboratory, one strain of bacteria can double in number every 15 minutes. Suppose a culture starts with 60 cells, use your graphing calculator or a table of values to show the sample’s growth after 2 hours.

We know that this relationship can be modeled with an exponential function since the bacteria is doubling. This means that it can be written as an equation in the form y=abx, where $a$ represents the initial value and $b$ represents the rate of change.

$y = 60 \times 2^{x}$ represents the growth rate of the bacteria where y is the number of cells there are after x quarter hours.

Create a table to go with the equation of the function.

x	y=60×2x	y
0	y=60×20	60
1	y=60×21	120
2	y=60×22	240
3	y=60×23	480
4	b=60×24	960
5	b=60×25	1920
6	b=60×26	3840
7	b=60×27	7680
8	b=60×28	15360

Graph the function.

y=60(2)^x..PNG

Example 2

The rate of change for this exponential function is 23 and the y-intercept (initial value) is 2. Write an equation, create a table, and create a graph to model this function.

First write the function in the f form $y = a b^{x}$ , where a represents the starting point and b represents the rate of change.

$y = 2 \times {(\frac{2}{3})}^{x}$

Then, fill in the table of values.

x	-3	-2	-1	0	1	2	3
y	$\frac{27}{4}$	$\frac{9}{2}$	$3$	$2$	$\frac{4}{3}$	$\frac{8}{9}$	$\frac{16}{27}$

Next, graph the function.

y=2(.67)^x.PNG

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