Unit 5 Day 1 – Exponential Functions – Introduction
I am so sorry that I have to be out today! I hope you all had a great break! Please work
through this packet today and I will be back tomorrow. –Mrs. H
Exponential Functions
An Exponential Function is a function of the form
f(x)=abx
coefficient
where a ≠ 0 and b > 0.
a is called the coefficient
b is called the base
variable is in the
exponent
base
Examples: f(x) = 3(2x), F(x) = (½)x
Note: the base is a constant and the exponent is the variable.
Given the following exponential functions, identify the coefficient and the base:
f(x) = -3(2)x
coefficient: _________
base: ____________
f(x) = 5(4)x
coefficient: _________
base: ____________
Use the following tables and coordinate grid to graph each function. Graph each of them in a
different color on the same graph.
x
x
x
2
x
-2
-1
0
1
2
3
f(x)
1
4
1
2
x
-2
-1
0
1
2
4
f(x)
1
9
1
3
x
-2
-1
0
1
2
f(x)
1
16
1
4
What is the domain of these functions? __________________________
What is the range of these functions? ___________________________
Did you notice an asymptote? Is so, what kind and where is it? ________________________
_______________________________________________________________________
Do the functions have an x-intercept? ________________
Do the functions have a y-intercept? _________________ If so, what is it? ____________
Notice that the graph of 2x contains the points (0, 1), (1, 2), and (-1, 1/2); the graph of 3x
contains the points (0, 1), (1, 3), and (-1, 1/3); and the graph of 4x contains the points (0, 1), (1,
4), and (-1, 1/4). That leads to this fact:
the graph of bx contains the points (0, 1), (1, b), and (-1, 1/b)
•
Given this fact list 3 points that the graph of 7x would contain: _______________________
Use the following tables and coordinate grid to graph each function. Graph each of them in a
different color on the same graph.
1
 
2
x
1
 
3
x
x
f(x)
x
f(x)
-2
-1
0
1
2
4
2
-2
-1
0
1
2
9
3
1
 
4
x
x
f(x)
-2 16
-1 4
0
1
2
What is the domain of these functions? __________________________
What is the range of these functions? ___________________________
Did you notice an asymptote? Is so, what kind and where is it? ________________________
_______________________________________________________________________
Do the functions have an x-intercept? ________________
Do the functions have a y-intercept? _________________ If so, what is it? ____________
x
x
1
1
Notice that the graph of   contains the points (0, 1), (1, ½), and (-1, 2); the graph of  
2
3
1
contains the points (0, 1), (1, 1/3), and (-1, 3); and the graph of   contains the points (0, 1),
4
(1, 1/4), and (-1, 4). That leads to this fact:
•
the graph of bx contains the points (0, 1), (1, b), and (-1, 1/b)
x
1
Given this fact list 3 points that the graph of   would contain: ____________________
8
The exponential function y = abx represents a quantity changing at a constant ratio.
If the ratio of the consecutive y-values is constant, then the data represents an exponential
function.
That means if I divide consecutive y-values and keep getting the same number, I have an
exponential function. Look at the following table:
x
f(x)
-1
3
0
6
1
12
2
24
3
48
x
f(x)
-1
3
0
6
1
12
2
18
3
30
This function is exponential because dividing each
y –value by the previous y-value gives me 2 every
time.
This function is NOT exponential because dividing
each y –value by the previous y-value does not give
me the same number every time.
Given a function with the following values, determine if it is an exponential function:
x
f(x)
x
f(x)
x
f(x)
0
2
0
2
0
3
1
4
1
4
1
9
2
8
2
6
2
27
3
16
3
8
3
81
4
32
4
10
4
243
Notice that when there is a common ratio, that the common ratio is the base of the exponential
function. Now we need to find a way to determine what the coefficient is. Look at the values
for each exponential function below:
y = 3(2)x
y = 4(2)2
y = 4(3)x
x
f(x)
x
f(x)
x
f(x)
0
3
0
4
0
4
1
6
1
8
1
12
2
12
2
16
2
36
3
24
3
32
3
108
4
48
4
64
4
324
base = 2
coefficient = 3
(value at x = 0)
base = 2
coefficient = 4
(value at x = 0)
base = 3
coefficient = 4
(value at x = 0)
So, the base of an exponential function is equal to the common ratio and the coefficient is equal
to the function’s value when x = 0.