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Section 10.3
Graphing Exponential
Functions
Copyright © 2011 Pearson Education, Inc.
Graphing Exponential Functions with b > 1
Graphing Exponential Functions
Example
x
f
x

2
Graph  
by hand.
Solution
• List input–output pairs
(see table)
• Input increases by 1 and
output multiplies by 2
• Plot these points (see next
slide)
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 2
Graphing Exponential Functions with b > 1
Graphing Exponential Functions
Solution Continued
• Use graphing calculator
to verify
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 3
Graphing Exponential Functions with 0< b < 1
Graphing Exponential Functions
Example
x
1

Graph g  x   4   by hand.
 2
Solution
• List input–output pairs (see table)
• For example
• (–1, 8) is a solution
• x increases by 1, y is multiplied by ½
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 4
Graphing Exponential Functions with 0 < b < 1
Graphing Exponential Functions
Solution Continued
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 5
Base Multiplier Property; Increase or Decreasing Property
Base Multiplier Property
Property
For an exponential function of the form y = abx, if the
value of the independent variable increases by 1, the
value of the dependent variable is multiplied by b.
Illustration
•For the function f  x   2  3 , as the value of x
increases by 1, the value of y is multiplied by 3
x
3

•For the function f  x   5   , as the value of x
4
increases by 1, the value of y is multiplied by 3/4
x
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 6
Increase or Decrease Property
Base Multiplier Property
Property
Let f  x   a  b  , where a > 0. Then
• If b > 1, then the function f is increasing. We say
that the function grows exponentially (left).
• If 0 < b < 1, then the function f is decreasing. We
say that the function decays exponentially (right).
x
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 7
y-intercept of an Exponential Function
Intercepts
Property
For an exponential function of the form
x
y  a b 
the y-intercept is (0, a).
Illustration
• The function f  x   5  8 , the y-intercept is (0, 5)
x
1

• The function f  x   4   , the y-intercept is (0, 4)
7
x
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 8
Intercepts and Graph of an Exponential Function
Intercepts
Warning
Exponential function of the form y   b  , the yx
x
intercept is not (0, b). By writing y   b   1b , we
see that the y-intercept is (0, 1).
For example, for y  2 x , the y-intercept is (0, 1).
x
Example
x
1

Let f  x   6  
 2
1. Find the y-intercept of f.
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 9
Intercepts and Graph of an Exponential Function
Intercepts
Solution
x
1
x

• f  x   6   is of the form f  x   a  b  ,
 2
• We know that the y-intercept is (0, a), or (0, 6).
Example
2. Find the x-intercept of f.
Solution
• By base multiplier property, x increases by 1, y
value multiplies by ½
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 10
Intercepts and Graph of an Exponential Function
Intercepts
Solution Continued
• No number of halvings will
result in zero
• As x grows large, y gets
closer to the x-axis
• Called horizontal
asymptote
Example
3. Graph f by hand.
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 11
Intercepts and Graph of an Exponential Function
Intercepts
Solution
• Plot solutions from the table
• Verify on graphing
calculator
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 12
Graphs of Functions of the Form y=abx and y= –abx
Reflection Property
Example
1. Sketch and compare
the graphs of f  x   5  3
x
f  x   5  3 .
x
Solution
• Input–output pairs are
listed in the table
• g is a reflection of f across
the x-axis
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 13
Graphs of Functions of the Form y=abx and y= –abx
Reflection Property
Example
2. Find the domain and
range of f.
Solution
•
•
•
x
5(3) is defined for any
real number x
Domain: All real numbers
Range: All positive real
numbers
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 14
Graphs of Functions of the Form y=abx and y= –abx
Reflection Property
Example
2. Find the domain and
range of g.
Solution
•
•
•
x
– 5(3) is defined for any
real number x
Domain: All real numbers
Range: All negative real
numbers
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 15
Reflection Property
Reflection Property
Property
x
x
f
x


ab
and
g
x

ab




The graphs of
are
reflections of each other across the x-axis.
Illustration
• For all exponential functions the x-axis is a
horizontal asymptote
• The range of an exponential function f(x) = abx is
the set of all positive real numbers if a > 0, and
the range is the set of all negative real number if
a< 0.
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 16
Reflection Property
Reflection Property
Continued
• b > 1 (left) and 0 < b < 1 (right)
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 17
Finding Values of a Function from Its Graph
Reflection Property
Example
The graph of an exponential
function f is shown.
1. Find f(2).
Solution
• Blue arrow shows input of
x = 2 leads to an output
y=8
• f(2) = 8
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 18
Finding Values of a Function from Its Graph
Reflection Property
Example
2. Find x when f(x) = 2.
Solution
• Red arrow shows output of
y = 2 leads to an input x = -2
• x = –2 when f(x) = 2
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 19
Finding Values of a Function from Its Graph
Reflection Property
Example
3. Find x when f(x) = 0.
Solution
• Graphs of exponential
functions get close to zero,
but never reaches x-axis
• No value of x where
f(x) = 0
Copyright © 2011 Pearson Education, Inc.
Section 10.3
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 20