7.6 EXPONENTIAL FUNCTIONS:
Function Rule: An equation that describes
a function.
Exponent: A number that shows repeated
multiplication.
GOAL:
Definition:
An EXPONENTIAL FUNCTION is a function
of the form:
𝑦 =𝑎∙𝑏
Constant
Base
𝑥
Exponent
Where a ≠ 0, b > o, b ≠ 1,
and x is a real number.
IDENTIFYING: We must be able to
identify exponential functions from
given data values.
Ex:
Does the table represent an exponential
function? If so, provide the function rule.
x
0
1
2
3
y
-1
-3
-9
-27
To answer the question we must take a look
at what is happening in the table.
+1
+1
+1
(x)
0
1
2
3
(y)
-1
-3
-9
-27
×3
×3
×3
The starting point is -1 when x = 0
The independent variable x increases by 1
The dependent variable y is multiplied by 3
Taking the info to consideration, we can see
that the equation for the problem is:
+1
(x)
(y)
0
-1
+1
1
-3
×3
+1
2
-9
×3
3
- 27
x
y=a∙b
×3
Here the difference of ×3 becomes the base.
Notice: we begin with -1 when x = 0 or a = -1
x
y=a∙b
y=
x
-1∙3
YOU TRY IT:
Does the table represent an exponential
function? If so, provide the function
rule.
x
1
2
3
4
y
2
8
32
128
SOLUTION: Taking the info to consideration,
we can see that the equation for the problem
+1
+1
+1
is:
(x)
(y)
1
2
2
8
×4
3
32
×4
4
128
x
y=a∙b
×4
Here the difference of ×4 becomes the base.
Notice: we begin with 2 when x = 1 or a = 1/2
x
y=a∙b
 y =½
x
∙4
Summary:
Linear Functions:  y = mx + b
The difference in the independent
variable (y) is in form of addition or
subtraction.
Exponential Equations:  y = abx
The difference in the independent
variable (y) is multiplication
EVALUATING: We must be able to
evaluate exponential functions.
Ex:
An investment of $5000 doubles in
value every decade. Write a function
and provide the worth of the
investment after 30 years.
EVALUATING: To provide the solution
we must know the following
formula:
A=
x
P∙2
A = total
P = Principal (starting amount)
2 = doubles
x = time
SOLUTION:
An investment of $5000 doubles in value every
decade. Write a function and provide the worth
of the investment after 30 years.
Amount:
unknown
Principal: $5000
Doubles: 2
Time (x): 30 yrs
(3 decades)
x
P∙2
A=
3
A = 5000∙2
A = 5000∙(8)
A = 40,000
YOU TRY IT:
Suppose 30 flour beetles are left
undisturbed in a warehouse bin. The
beetle population doubles each week.
Provide a function and the population
after 56 days.
SOLUTION:
Suppose 30 flour beetles are left undisturbed in a
warehouse bin. The beetle population doubles
each week. Provide a function and the population
after 56 days.
Amount: unknown
Principal: 30
Doubles: 2
Time (x): 56 days
(8 weeks)
x
P∙2
A=
8
A = 30∙2
A = 30∙(256)
A = 7,680
GRAPHING: To provide the graph of the
equation we can go back to basics and
create a table.
Ex:
What is the graph of y = 3∙2x?
GRAPHING:
X
y = 3∙2x
y
-2
3∙2(-2)
-1
3∙2(-1) =
𝟑
𝟐𝟐
𝟑
𝟐𝟏
0
3∙2(0) = 3∙1
𝟑
𝟒
𝟑
𝟐
3
1
3∙2(1) = 3∙2
6
2
3∙2(2) = 3∙4
12
=
GRAPHING:
X
y
0
𝟑
𝟒
𝟑
𝟐
3
1
6
2
12
-2
-1
This graph grows fast = Exponential Growth
YOU TRY IT:
Ex:
What is the graph of y = 3∙
𝟏 x
?
𝟐
GRAPHING:
X
-2
-1
0
1
2
y = 3∙
𝟏 x
𝟐
y
3∙
𝟏 (-2)
2
=3∙(2)
𝟐
12
3∙
𝟏 (-1)
1
=3∙(2)
𝟐
6
3∙
𝟏 (0)
𝟐
3
3∙
𝟏 (1)
𝟏
=3∙
𝟐
𝟐
3∙
𝟏 (2)
𝟏
=3∙
𝟐
𝟒
= 3∙1
𝟑
𝟐
𝟑
𝟒
GRAPHING:
X
y
-2
12
-1
6
0
3
1
2
𝟑
𝟐
𝟑
𝟒
This graph goes down = Exponential Decay
VIDEOS:
Exponential
Functions
Growth
https://www.khanacademy.org/math/trigonometry/expon
ential_and_logarithmic_func/exp_growth_decay/v/expone
ntial-growth-functions
Graphing
https://www.khanacademy.org/math/trigonometry/expon
ential_and_logarithmic_func/exp_growth_decay/v/graphi
ng-exponential-functions
VIDEOS:
Exponential
Functions
Decay
https://www.khanacademy.org/math/trigonometry/expon
ential_and_logarithmic_func/exp_growth_decay/v/wordproblem-solving--exponential-growth-and-decay
CLASSWORK:
Page 450-451:
Problems: As many as needed
to master the
concept.