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Choices…
• A wealthy man nearing the end of his life called
his two children to his bedside. He wanted to
leave them with the opportunity to experience the
richness of life that he had enjoyed.
• He offered them the choice of $1,000,000.00
cash or
• $.01 cash (yes, ONE Penny) that would double
everyday for one month (30 days). He then sent
them home to consider the offer.
• Which choice is better? Why?
The Answer…
Remember, you
still have to add
up everything
for each day…
9.5 Notes
I. Exploring Exponential Functions.
Let’s examine exponential functions. They are
different than any of the other types of functions we’ve
studied because the independent variable is in the
exponent.
Let’s look at the graph of
this function by plotting
x
x
2
x some points.
8
3
2
1
0
-1
-2
-3
8
4
2
1
1/2
1/4
1/8
f x   2
BASE
Recall what a
negative exponent
means:
1
1


f 1  2 
2
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
The box summarizes the general shapes of exponential function
graphs.
Graphs of Exponential Functions
Compare the graphs 2x, 3x , and 4x
Characteristics about the
Graph of an Exponential
x
Function f x   a where a > 1
1. Domain is all real numbers
2. Range is positive real numbers
3. There are no x intercepts because
there is no x value that you can put
in the function to make it = 0
4. The y intercept is always (0,1)
because a 0 = 1
5. The graph is always increasing
6. The x-axis (where y = 0) is a
horizontal asymptote for x  - 
f x   4 x
f  x   3x
f x   2 x
Can
What
What
you
is
isthe
the
seerange
x
What
Are
these
is
the
ythe
of
intercept
horizontal
domain
an exponential
of
of
these
intercept
exponential
ofan
these
function?
exponential
asymptote
exponential
for
exponential
functions
functions?
these
function?
functions?
functions?
increasing
or
decreasing?
II. Graphing Exponential Functions with a Table
Ex 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.
Ex 2: 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.
•
•
•
•
Ex 3: Graph y = 0.2(5)x.
Choose several values of x
and generate ordered pairs.
x
–1
0
1
2
Graph the ordered pairs
and connect with a smooth
curve.
•
0.2(5)x
y=
0.04
0.2
1
5
• •
•
Ex 4:
Choose several values of x
and generate ordered pairs.
1 x
x y =– (2)
4
–1
0
1
2
–0.125
–0.25
–0.5
–1
Graph the ordered pairs
and connect with a smooth
curve.
• • •
•
Ex 5:
Graph y = –6x.
Choose several values of x
and generate ordered pairs.
x
–1
0
1
2
y = –6x
–0.167
–1
–6
–36
Graph the ordered pairs
and connect with a smooth
curve.
•
•
•
Ex 6:
Graph y = –3(3)x.
Choose several values of x
and generate ordered pairs.
x
–1
0
1
2
y = –3(3)x
–1
–3
–9
–27
Graph the ordered pairs
and connect with a smooth
curve.
•
•
•
Ex 7:
Graph each exponential function.
Choose several values of x
and generate ordered pairs.
x
–1
0
1
2
Graph the ordered pairs
and connect with a smooth
curve.
1
y = –1( )x 4
–4
–1
–0.25
– 0.0625
•
•
• •
Ex 8:
Graph each exponential function.
y = 4(0.6)x
Choose several values of x
and generate ordered pairs.
x
–1
0
1
2
y = 4(0.6)x
6.67
4
2.4
1.44
Graph the ordered pairs and
connect with a smooth
curve.
•
•
•
•
Ex 9:
Graph each exponential function.
Choose several values of x
and generate ordered pairs.
x
–1
0
1
2
Graph the ordered pairs
and connect with a smooth
curve.
x1
y = 4( ) 4
16
4
1
0.25
•
•
•
•
Ex 10:
Graph each exponential function.
y = –2(0.1)x
Choose several values of x
and generate ordered pairs.
x
–1
0
1
2
y = –2(0.1)x
–20
–2
–0.2
–0.02
Graph the ordered pairs
and connect with a smooth
curve.
•
•
•
•
II. The Base “e” (also called the natural base)
To model things in nature, we’ll
need a base that turns out to be
between 2 and 3. Your calculator
knows this base. Ask your
calculator to find e1. You do this by
using the ex button (generally you’ll
need to hit the 2nd or yellow button
first to get it depending on the
calculator). After hitting the ex, you
then enter the exponent you want
(in this case 1) and push = or enter.
If you have a scientific calculator
that doesn’t graph you may have to
enter the 1 before hitting the ex. Example
You should get 2.718281828
for TI-83
f x   e x
f  x   3x
f x   2 x
III. Identifying Exponential Functions From a Table
Exponential functions 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.
y
Ex 1:
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
+1
constant amount.
+1
+1
x
0
1
2
3
y
4
12
36
108
3
3
3
Ex 2:
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 x-values increase by a
constant amount, the y-values are
not multiplied by a constant
amount.
+1
+1
+1
x
y
–1 –64
0
0
1 64
2 128
+ 64
+ 64
+ 64
Ex 3:
Tell whether each set of ordered pairs satisfies an exponential
function. Explain your answer.
{(–1, 1), (0, 0), (1, 1), (2, 4)}
This is not an exponential function.
As the x-values increase by a
constant amount, the y-values are
not multiplied by a constant
amount.
+1
+1
+1
x
–1
0
1
2
y
1
0
1
4
–1
+1
+3
Ex 4:
Tell whether each set of ordered pairs satisfies an exponential
function. Explain your answer.
{(–2, 4), (–1 , 2), (0, 1), (1, 0.5)}
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
y
–2 4
–1 2
0
1
1 0.5
× 0.5
× 0.5
× 0.5
IV. Applications
Ex 1: In 2000, each person in India consumed an average of 13
kg of sugar. Sugar consumption in India is projected to increase
by 3.6% per year. At this growth rate the function
f(x) = 13(1.036)x gives the average yearly amount of sugar, in
kilograms, consumed per person x years after 2000. Using this
model, in about what year will sugar consumption average about
18 kg per person?
Enter the function into the
Y = editor of a graphing
calculator.
Press
. Use the arrow
keys to find a y-value as
close to 18 as possible. The
corresponding x-value is 9.
The average consumption will reach 18 kg in 2009.
Ex 2: An accountant uses f(x) = 12,330(0.869)x, where x is the time
in years since the purchase, to model the value of a car. When will
the car be worth $2000?
Enter the function into the Y =
editor of a graphing calculator.
Check It Out! Example 6 Continued
An accountant uses f(x) = 12,330(0.869)x, is the time in years since
the purchase, to model the value of a car. When will the car be
worth $2000?
Press
. Use the arrow
keys to find a y-value as
close to 2000 as possible.
The corresponding x-value is
13.
The value of the car will reach $2000 after about year 13.
Lesson Quiz: Part I
Tell whether each set of ordered pairs satisfies an exponential
function. Explain your answer.
1. {(0, 0), (1, –2), (2, –16), (3, –54)}
No; for a constant change in x, y is not multiplied by the same
value.
2. {(0,–5), (1, –2.5), (2, –1.25), (3, –0.625)}
Yes; for a constant change in x, y is multiplied by the same
value.
Lesson Quiz: Part II
3. Graph y = –0.5(3)x.
Lesson Quiz: Part III
4. The function y = 11.6(1.009)x models residential
energy consumption in quadrillion Btu where x is
the number of years after 2003. What will
residential energy consumption be in 2013?
 12.7 quadrillion Btu
5. In 2000, the population of Texas was about 21
million, and it was growing by about 2% per
year. At this growth rate, the function
f(x) = 21(1.02)x gives the population, in
millions, x years after 2000. Using this model,
in about what year will the population reach 30
million? 2018