3.2 Graph Exponential Decay Functions

P. 236What is exponential decay?

How can you recognize exponential growth and decay from the equation?

What is the form of a general exponential decay equation?

What is the real life equation for exponential decay?

Exponential Decay

• Has the same form as growth functions f(x) = abx

• Where a > 0

• BUT:

• 0 < b < 1 (a fraction between 0 & 1)

Recognizing growth and decay functions

• State whether f(x) is an exponential growth or decay function

• f(x) = 5(2/3)x

• b=2/3, 0<b<1 it is a decay function.

• f(x) = 8(3/2)x

• b= 3/2, b>1 it is a growth function.

• f(x) = 10(3)-x

• rewrite as f(x)=10(1/3)x so it is decay

Recall from 4.1:

• The graph of y= abx • Passes thru the point (0,a) (the y

intercept is a)• The x-axis is the asymptote of the

graph• a tells you up or down• D is all reals (the Domain)• R is y>0 if a>0 and y<0 if a<0 • (the Range)

Graph:• y = 3(1/4)x

• Plot (0,3) and (1,3/4)

• Draw & label asymptote

• Connect the dots using the asymptote

Domain = all reals Range = reals>0

y=0

Graph y =12

x

SOLUTION

STEP 1 Make a table of values

STEP 2 Plot the points from the table.

STEP 3 Draw, from right to left, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the left.

Graph• y = -5(2/3)x

• Plot (0,-5) and (1,-10/3)

• Draw & label asymptote

• Connect the dots using the asymptote

y=0

Domain : all realsRange : y < 0

Now remember: To graph a general Exponential

Function:• y = a bx-h + k

• Find your asymptote from k

• Pick values for x. Try to make your exponent value 0 or 1.

• Complete your T chart (Find y).

• Sketch the graph.

Example graph y=-3(1/2)x+2+1• h=-2, k=1• Asymptote y = 1• Pick values for x

that make the value of the exponent 0 or 1

x y

−2

−1

−2

−½

y=1

Domain : all realsRange : y<1

Using Exponential Decay Models

• When a real life quantity decreases by fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by:

• y = a(1-r)t

• Where aa is the initial amount and rr is the percent decrease expressed as a decimal.

• The quantity 1-r is called the decay factor

Ex: Buying a car!• You buy a new car for $24,000. • The value y of this car decreases

by 16% each year.• Write an exponential decay model

for the value of the car.• Use the model to estimate the

value after 2 years.• Graph the model.• Use the graph to estimate when

the car will have a value of $12,000.

• Let t be the number of years since you bought the car.

• The model is: y = a(1-r)t

• = 24,000(1-.16)t

• = 24,000(.84)t

• Note: .84 is the decay factor

• When t = 2 the value is y=24,000(.84)2 ≈ $16,934

Now Graph

The car will have a value of $12,000 in 4 years!!!

• What is exponential decay?As the values of x increases, the values of the

function decreases.• How can you recognize exponential growth

and decay from the equation?Growth is greater than 1 and decay is greater

than 0 and less than 1.• What is the form of a general exponential

decay equation?y = abx-h +k• What is the real life equation for exponential

decay?y = a(1-r)t

3.2 Assignment!

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