Chabot Mathematics
§9.1a
Exponential Fcns
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Review § 8.5
MTH 55
 Any QUESTIONS About
• §8.5 → Rational InEqualities
 Any QUESTIONS About HomeWork
• §8.5 → HW-41
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Exponential Function
 A function, f(x), of the form
f x   a ,
x
a  0 and a  1,
 is called an EXPONENTIAL function with
BASE a.
 The domain of the exponential function is
(−∞, ∞); i.e., ALL Real Numbers
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Recall Rules of Exponents
 Let a, b, x, and y be real numbers with
a > 0 and b > 0. Then
a a  a
x
y
xy
a   a
x y
,
x
a
xy
a ,
y
a
ab 
x
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a b ,
x
x
xy
,
a  1,
0
x
a
x
1  1
 x   .
 a
a
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Evaluate Exponential Functions
 Example 
a. Let f x   3x2. Find f 4 .
 Solution a. f 4   342  32  9
 Example b.
 Let g x   2 10 . Find g 2 .
1
1
2
b. g 2   2 10  2  2  2 
 Solution 
10
100
1
1
2
b. g 2   2 10  2  2  2 
 0.02
10
100
x
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Evaluate Exponential Functions
x
 1
 3
 Example c.
 Let h x     . Find h    .
 9
 2
 3  1 
 Solution

c. Let h      
 2 3  9 

 3  1  2
1
c. Let h        9
 2  9

3
2
 
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
3
2
3

1 2
 
 9
3
2
 9  27
3
2
 9  27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Graph y = f(x) =3x
 Graph the exponential fcn: f ( x)  3x.
 Make T-Table,
& Connect Dots
x
y
0
1
–1
2
–2
3
1
3
1/3
9
1/9
27
Chabot College Mathematics
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y
8
7
6
5
4
3
2
1
-5 -4 -3 -2 -1
1
2 3 4 5
-1
x
-2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Graph Exponential
x
1

 Graph the exponential fcn: f ( x)    .
 3
 Make T-Table,
y
y  3x
8
& Connect Dots
7
x
y
0
1
–1
1
1/3
3
2
–2
–3
1/9
9
27
Chabot College Mathematics
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1
y  f ( x)   
 3
x
6
5
4
3
2
1
-5 -4 -3 -2 -1
1
2 3 4 5
-1
x
-2
• This fcn is a
REFLECTION of y = 3x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Graph Exponential
x
 1
 Graph the exponential fcn: y    .
 2
 Construct SideWays T-Table
x
−3 −2 −1 0
y = (1/2)x 8
4
2
1
1
2
3
1/2
1/4
1/8
 Plot Points and Connect Dots with
Smooth Curve
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Graph Exponential
 As x increases in the positive direction,
y decreases towards 0
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Exponential Fcn Properties
 Let f(x) = ax, a > 0, a ≠ 1. Then
A. The domain of f(x) = ax is (−∞, ∞).
B. The range of f(x) = ax is (0, ∞); thus,
the entire graph lies above the x-axis.
C. For a > 1 (e.g., 7)
i.
f is an INcreasing function; thus, the graph
is RISING as we move from left to right
ii. As x→∞, y = ax increases indefinitely
and VERY rapidly
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Exponential Fcn Properties
Let f(x) = ax, a > 1, a ≠ 1. Then

iii. As x→−∞, the values of y = ax get
closer and closer to 0.
D. For 0 < a < 1 (e.g., 1/5)
i.
f is a DEcreasing function; thus, the graph
is falling as we scan from left to right.
ii. As x→−∞, y = ax increases indefinitely and
VERY rapidly
iii. As x→ ∞, the values of y = ax get closer
and closer to 0
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Exponential Fcn Properties
 Let f(x) = ax, a > 0, a ≠ 1. Then
E. Each exponential function f is
one-to-one; i.e., each value of x has
exactly ONE target. Thus:
i.
a a
x1
x2
 x1  x2
ii. f has an inverse
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Exponential Fcn Properties
 Let f(x) = ax, a > 0, a ≠ 1. Then
F. The graph f(x) = ax has no x-intercepts
•
In other words, the graph of f(x) = ax
never crosses the x-axis. Put another
way, there is no value of x that will cause
f(x) = ax to equal 0
G. The x-axis is a horizontal asymptote
for every exponential function of the
form f(x) = ax.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Translate Exponential Graphs
Translation
Equation
Effect on Equation
Horizontal
Shift
y = ax+b
= f (x + b)
Shift the graph of
y = ax, b units
(i) Left if b > 0.
(ii) Right if b < 0.
Vertical
Shift
y = ax + b
= f (x) + b
Shift the graph of
y = ax, b units
(i) Up if b > 0.
(ii) Down if b < 0.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Sketch
x
a
.
f
x

3
4


Graph
 By Translation
Move DOWN
y = 3x by
3 Units
 Note
• Domain: (−∞, ∞)
• Range: (−4, ∞)
• Horizontal
Asymptote:
y = −4
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
b. f x   3
Example  Sketch Graph
x1
 By Translation
Move LEFT
y = 3x by
1 Unit
 Note
• Domain: (−∞, ∞)
• Range: (0, ∞)
• Horizontal
Asymptote:
y=0
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Alternative Graph: Swap x & y
 It will be helpful in later work to be able
to graph an equation in which the x
and y in y = ax are interchanged
12
y  2.3x
10
8
6
4
x  2.3
2
0
y
-2
-4
-6
-6
Chabot College Mathematics
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-4
-2
0
2
4
6
8
10
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Graph x = 3y
y
x3 .
 Graph the exponential fcn:
 Make T-Table,
& Connect Dots
Chabot College Mathematics
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y
6
5
4
x
y
3
2
1
3
1/3
0
1
–1
1
9
1/9
27
2
–2
3
-3 -2 -1
-1
-2
1
2 3 4 5
6 7 8 9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
x
Example  Apply Exponential
 Example  Bank Interest compounded
annually.
 The amount of money A that a principal
P will be worth after t years at interest
rate i, compounded annually, is given by
the formula
t
A  P(1  i ) .
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Compound Interest

Suppose that $60,000 is invested at
5% interest, compounded annually
a) Find a function for the amount in the
account after t years

SOLUTION
a)a) A(t )  P(1  i)t
= $60000(1 + 0.05 )t
= $60000(1.05)t
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Compound Interest

Suppose that $60,000 is invested at
5% interest, compounded annually
b) Find the amount of money amount in the
account at t = 6.

SOLUTION
b) A(6) = $60000(1.05)6  $80,405.74
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Bacterial Growth
 A technician to the Great French
microbiologist Louis Pasteur noticed
that a certain culture of bacteria in milk
doubled every hour.
 Assume that the bacteria count B(t) is
modeled by the equation
B t   2000  2 ,
t
• Where t is time in hours
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Bacterial Growth
 Given Bacterial
Growth Equation

B t   2000  2 ,
t
Find:
a) the initial number of bacteria,
b) the number of bacteria after 10 hours; and
c) the time when the number of bacteria will
be 32,000.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Bacterial Growth
a) INITIALLY time, t, is ZERO → Sub
t = 0 into Growth Eqn:
B0  B 0   2000  2  2000 1  2000
0
b) At Ten Hours Sub t = 10 into Eqn:
b. B 10   2000  2  2, 048, 000
10
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Bacterial Growth
c) Find t when B(t) = 32,000
32000  2000  2
16  2

2 2
4t
4
t
Thus 4 hours after the starting time,
the number of bacteria will be 32k
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t
t
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
WhiteBoard Work
 Problems From §9.1 Exercise Set
• 36, 40, 54
 USA
Personal
Savings
Rate
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
All Done for Today
Bacteria
Grow
FAST!
•
Note: 37 °C = 98.6 °F (Body Temperature)
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Irrational Exponents
 By The Properties of Exponents we Can
Evaluate Bases Raised to
Rational-Number Powers Such as
2
3
 
7  7
1
2 3

 7   7

3
2
1
3
2

 


 7
3
2
 What about expressions with
2
IRrational exponents such as: 7 ,
 To attach meaning to this expression
consider a rational approximation,
r, for the Square Root of 2
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Irrational Exponents
 Approximate by
ITERATION on:
p  7r  7
2
2
r closes in on 2
p closes in on 7
1.4 < r < 1.5
15.245  71.4  p  71.5  18.520
1.41 < r < 1.42
15.545  71.41  p  71.42  18.850
1.414 < r < 1.415
15.666  71.414  p  71.415  15.697
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r 2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Irrational Exponents
 Thus by Iteration
2
7
 15.6728909
 Any positive irrational exponent can be
interpreted in a similar way.
 Negative irrational exponents are then
defined using reciprocals.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt