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 ∙𝑏𝑥

BaseExponentWhere a ≠ 0, b > o, b ≠ 1,

and x is a real number.

Constant

IDENTIFYING: We must be able to identify exponential functions from given data values.

x 0 1 2 3

y -1 -3 -9 -27

Ex: Does the table represent an exponential function? If so, provide the function rule.

To answer the question we must take a look at what is happening in the table.

(x) 0 1 2 3

(y) -1 -3 -9 -27

+ 1 + 1 + 1

×3 ×3 ×3

The dependent variable y is multiplied by 3The independent variable x increases by 1The starting point is -1 when x = 0

Taking the info to consideration, we can see that the equation for the problem is:

(x) 0 1 2 3(y) -1 - 3 - 9 - 27

+ 1 + 1 + 1

×3 ×3 ×3

y = -1 3∙ x

y=a b∙ x

Notice: we begin with -1 when x = 0 or a = -1Here the difference of ×3 becomes the base.

y=a b∙ x

YOU TRY IT:

x 1 2 3 4

y 2 8 32 128

Does the table represent an exponential

function? If so, provide the function rule.

Taking the info to consideration, we can see that the equation for the problem is:

SOLUTION:

y =½ 4∙ x

Notice: we begin with 2 when x = 1 or a = 1/2Here the difference of ×4 becomes the base.

y=a b∙ x

(x) 1 2 3 4(y) 2 8 32 128

+ 1 + 1 + 1

×4 ×4 ×4

y=a b∙ x

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 = P∙2x

A = totalP = Principal (starting amount) 2 = doublesx = time

SOLUTION:

Amount:

An investment of $5000 doubles in value every decade. Write a function and provide the worth of the investment after 30 years.

$5000 Principal:

Doubles: 2

Time (x): 30 yrs (3 decades)

unknown A = P∙2x

A = 5000∙23

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:

Amount:

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.

30 Principal:

Doubles: 2

Time (x): 56 days (8 weeks)

unknown A = P∙2x

A = 30∙28

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 2∙ x?

GRAPHING:X y = 3 2∙ x y

-2 3 2∙ (-2) = 𝟑𝟒

3 2∙ (-1) = 𝟑𝟐-1

0 3 2∙ (0) 3 = 3 1 ∙

1 3 2∙ (1) 6 = 3 2 ∙

2 3 2∙ (2) 12 = 3 4 ∙

GRAPHING: X y-2 𝟑

𝟒𝟑𝟐-1

0 3

1 6

2 12

This graph grows fast = Exponential Growth

YOU TRY IT:

Ex: What is the graph of y = 3∙x?

GRAPHING:X y = 3∙x y

-2 3 ∙ (-2) 12

6-1

0 3 = 3 1 ∙

1

2

=3 (2)∙ 2

3 ∙ (-1) =3 (2)∙ 1

3 ∙ (0)

3 ∙ (1) =3 ∙ 𝟑𝟐𝟑𝟒

3 ∙ (2) =3 ∙

GRAPHING: X y-2

𝟑𝟒

𝟑𝟐

-1

0 3

1

6

2

12

This graph goes down = Exponential Decay

VIDEOS: ExponentialFunctions

Growthhttps://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/exponential-growth-functionsGraphing

https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/graphing-exponential-functions

VIDEOS:ExponentialFunctions

Decay

https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/word-problem-solving--exponential-growth-and-decay

CLASSWORK:

Page 450-451:

Problems: As many as needed to master the concept.