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12.2
Graphing Exponential Functions
12.2
A function rule that describes the pattern above is f(x) = 2(3)x. This type of
function, in which the independent variable appears in an exponent, is an
exponential function. Notice that 2 is the starting population and 3 is the amount
by which the population is multiplied each day.
12.2
Remember that linear functions have constant
first differences. Exponential functions do not
have constant differences, but they do have
constant ratios.
As the x-values increase by a constant amount, the y-values are multiplied by a
constant amount. This amount is the constant ratio and is the value of b in f(x) = abx.
12.2
Example 1: Identifying an Exponential Function
Tell whether each set of ordered pairs satisfies an exponential function. Explain
your answer.
{(0, 4), (1, 12), (2, 36), (3, 108)}
This is an exponential function. As the x-values
increase by a constant amount, the y-values
are multiplied by a constant amount.
+1
+1
+1
x
0
1
2
3
y
4
12
36
108
3
3
3
12.2
Example 2: Identifying an Exponential Function
Tell whether each set of ordered pairs satisfies an exponential function. Explain
your answer.
{(–1, –64), (0, 0), (1, 64), (2, 128)}
This is not an exponential function. As the xvalues increase by a constant amount, the yvalues are not multiplied by a constant amount.
+1
+1
+1
x
y
–1 –64
0
0
1 64
2 128
+ 64
+ 64
+ 64
12.2
Example 3
Graph y = 2x.
Choose several values of x and
generate ordered pairs.
x
–1
0
1
2
y = 2x
0.5
1
2
4
Graph the ordered pairs and
connect with a smooth curve.
•
•
•
•
12.2
Example 4: Graphing y = abx with a > 0 and b > 1
Graph y = 0.5(2)x.
Choose several values of x and
generate ordered pairs.
x y = 0.5(2)x
–1
0.25
0
0.5
1
1
2
2
Graph the ordered pairs and
connect with a smooth curve.
•
• •
•
12.2
Example 5: Graphing y = abx with a < 0 and b > 1
x
1
Graph y   
2

Choose several values of x and
generate ordered pairs.
X
-2
1
y 
2
Graph the ordered pairs and
connect with a smooth curve.
x
4
–1
2
0
1
1
0.5
2
0.25
•
•
•
•• •
12.2
Example 6
Graph y = –3(3)x +2
Choose several values of x and
generate ordered pairs.
x
–1
0
1
2
y=–
3(3)x+2
1
–1
–7
–25
Graph the ordered pairs and
connect with a smooth curve.
•
•
•
12.2
The box summarizes the general shapes of exponential function graphs.
Graphs of Exponential Functions
a>0
a>0
a<0
a<0
For y = abx, if b > 1, then the
graph will have one of these
shapes.
For y = abx, if 0 < b < 1, then the
graph will have one of these
shapes.
Graphing Exponential Equations
• For the our purposes we will be just
sketching exponential graphs.
– We will need several pieces of information in
order to graph the function
• The general shape of the graph
• Two basic points
• The asymptote of the function
General Shape Review
Growth
Reflected
Growth
Decay
Reflected
Decay
Finding two basic points
To find the points we are going to use a basic “t” table.
y  23
x
x
We need to find y….
0
We need to find y….
1
y
Hint:
Remember that anything to the 0 power is 1!
Asymptotes
• An asymptote is a line that the graph
approaches but never touches.
• For graphs that have no vertical shift, the
asymptote is y = 0.
• For graphs that have a vertical shift, the
asymptote is just shifted up or down as
well.
Let’s put it together…..
x
y

2

3
Graph:
1. We know its general
shape is…
2. We have our 2
basic points….
x
y
0
2
1
6
3. Our asymptote is y = 0
Let’s try one …
Graph:
1
y  2   
2
x
1. We know its general
shape is…
2. We have our 2
basic points….
x
y
0
1
3. Our asymptote is y = 0
PUT IT ALL TOGETHER AND YOU HAVE GRAPHED A N
EXPONENTIAL FUNCTION!
Let’s try one more…
Graph:
y  3  2  2
x
1. We know its general
shape is…
2. We have our 2
basic points….
x
y
0
1
3. Our asymptote is y = 2
PUT IT ALL TOGETHER AND YOU HAVE GRAPHED A N
EXPONENTIAL FUNCTION!