February 19, 2014
Section 3.5 ­ Exponential Functions
We've talked about:
Linear Functions, y = ax + b
a is slope (vertically stretches or shrinks graph)
b is y­intercept (or vertical shift)
Quadratic Functions, y = ax 2 + bx + c
In particular, we looked at y = ax 2 + c
a vertically stretches or shrinks graph
c vertically shifts graph
Square­Root Functions, y = a √(x ­ b) + c
a vertically stretches or shrinks graph
b horizontally shifts graph
c vertically shifts graph
We're now going to talk about exponential functions :
y = ab
x
What do you notice that's different between this and our other equations?
February 19, 2014
What does the graph of an exponential function look like?
y = 2
x
y
­2
­1
x
0
1
2
We're going to compare these graphs:
y = 2
x
y = 4
x
y = 0.5
x
February 19, 2014
What's happening?
Domain (possible x values):
Range (possible y values):
February 19, 2014
February 19, 2014
Compare graphs:
(Use your calculator)
y = 0.1x
y = 0.5x
y = 0.9x
What's happening?
Domain (possible x values):
Range (possible y values):
February 19, 2014
What's happening?
Domain (possible x values):
Range (possible y values):
y = ab
x
y = 2x
y = 4x
y = 0.5x
y = 0.1x
y = 0.5x
y = 0.9x
What is the a value in all the graphs we did? How do you know?
What can you say about how the graph will look based on the value of b?
February 19, 2014
Exponential functions are always increasing or always decreasing:
An increasing function is called exponential growth .
A decreasing function is called exponential decay.
Because the graph gets closer and closer to the x­axis without ever crossing it, we call the x­axis (or the line
y = 0) an asymptote .
Review:
What does the equation of an exponential function look like?
What does the graph of an exponential function look like?
What can you say about how the graph will look based on the value of b?
February 19, 2014
Compare graphs:
(Use your calculator)
y = (1)2
x
y = (3)2
x
y = (5)2
x
What's happening?
Domain (possible x values):
Range (possible y values):
February 19, 2014
y = ab
x
What is the b value in these graphs? How do you know?
y = (1)2x
y = (3)2x
y = (5)2x
What can you say about how the graph will look based on the value of a?
Conclusions:
For the exponential function, y = ab
x
The impact of a:
(0, a) is the y­intercept.
(a also vertically stretches or shrinks the graph.)
The impact of b:
If b > 1 , the exponential function is increasing or growing .
If 0 < b < 1 , the function is decreasing or decaying .
February 19, 2014
Finding exponential functions from a table:
In a linear function, equal steps in x produced equal steps in y .
In exponential functions, equal steps in x produce equal ratios in y .
KNOW THIS!
Note: Always calculate the common ratio by dividing a point by the previous point!
y = ab
x
The common ratio is b!
The y­intercept gives us the value of a!
y = 2
x
Make a table:
+1
x
­2
+1
­1
0
1
2
y 0.25 0.5 1 2 4
0.5 ÷ 0.25 = 2
4 ÷ 2 = 2
In exponential functions, equal steps in x produce equal ratios in y.
Always calculate the common ratio by dividing a point by the previous point!
February 19, 2014
y = 2
x
Make a table:
+1
x
­2
+1
­1
0
1
2
y 0.25 0.5 1 2 4
4 ÷ 2 = 2
0.5 ÷ 0.25 = 2
The common ratio is b. What is b?
The y­intercept gives us the value of a. What is a?
For problems 27 ­ 38, determine whether the table represents an exponential function. If it is exponential, give the common ratio and tell whether it is a growth or a decay function.
x
0
1
2
3
4
y 64 32 16 8 4
____ ÷ ____ = ____
What is the common ratio?
Is it growth or decay?
____ ÷ ____ = ____
February 19, 2014
For exercises 39 ­ 44, use the tables to find the equation of the exponential function. Identify whether the equation represents a growth or decay function.
x
­1
0
1 2 3
c(x) 80 8 0.8 0.08 0.008 ____ ÷ ____ = ____
____ ÷ ____ = ____
y = abx
The common ratio is b. What is the common ratio?
The y­intercept gives us the value of a. What is the y­
intercept?
What is the equation?
Is it growth or decay?
Find the equation representing the following exponential function:
x
­2
­1
0
1
2
y
7 1/9
5 1/3
4
3
2 1/4
If you are told it's an exponential function , choose points wisely to find the common ratio.
y = abx
The common ratio is b. What is the common ratio?
The y­intercept gives us the value of a. What is the y­intercept?
What is the equation?
Is this a growth or decay function? How can you tell?
February 19, 2014
Determine if the following is an exponential function
the equation if it is:
x
­2
­1
0
1
2
y
16
4
1/2
1/4
1/16
, then find In exponential functions, equal steps in x produce equal ratios in y. Is that happening here?
y = abx
The common ratio is b. What is the common ratio?
The y­intercept gives us the value of a. What is the y­intercept?
What is the equation?
What kind of functions are these?
(linear, quadratic, square root, exponential, none)
y = 2 x
y = x 2
y = 2x
y = 2 √x
y = 2/x
y = (3/2)(2/3)
y = x(x + 1)
3y + 2x = 0
x
y = (3/2)x
y = 4
2/3
y = √(3) + x