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Function Rule: An equation that describes a function. 7.6 EXPONENTIAL FUNCTIONS:. Exponent: A number that shows repeated multiplication. GOAL:. Definition:. An EXPONENTIAL FUNCTION is a function of the form:. Constant. Base. Exponent. - PowerPoint PPT Presentation
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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.
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