Exponential Functions

Table of Contents

Exponential Function

An exponential is a function in form of y=b^x, b>0 and $b\ne 1$

• has a repeating pattern of finite differences
• rate of change(increasing) that is proportional to the function for b>1
• rate of change(decreasing) that is proportional to the function for 0<b<1

Growth Example

Graph $2^x$

Table of Values

x	y
-2	1/4
-1	1/2
0	1
1	2
2	4

Graphing

Special Properties

	y=2^x
Domain	x ∈ ℝ
Range	y > 0
y-intercept	(0,1)
Asymptotes	x=0
Increasing or Decreasing	Increasing

Decay Example

graph $y=({\frac{1}{3}}^x)$

Table of values

x	y
-2	9
-1	3
0	1
1	1/3
2	1/9

Graphing

We are decaying

Decay And Growth Relation

Notice the similarity between $(3{)}^x$ and $(\frac{1}{3}{)}^x$ $(\frac{1}{3}{)}^x=((3{)}^{-1}{)}^x=(3{)}^{-x}$ Its a flip on the y axis

Growth Factor

to get the growth factor (or the base) from the table of values

Scale_differences

x	y	$\nabla y_1$
-2	$-\frac{1}{9}$
-1	$-\frac{1}{3}$	3
0	1	3
1	3	3
2	9	3
If you find the scale from one value of y to the other, and they are all the same, you will find the base.