holt algebra 2 7-1 exponential functions, growth, and decay holt algebra 2 read each slide. answer...

Holt Algebra 2

7-1 Exponential Functions, Growth, and Decay

Holt Algebra 2

Read each slide. Answer the hidden questions.

Evaluate.

1. 100(1.08)20

2. 100(0.95)25

3. 100(1 – 0.02)10

4. 100(1 + 0.08)–10

Holt Algebra 2


Growth that doubles every year can be modeled by using a function with a variable as an exponent. This function is known as an exponential function. The parent exponential function is f(x) = bx, where the base b is a constant and the exponent x is the independent variable.

Holt Algebra 2


The graph of the parent function f(x) = 2x is shown. The domain is all real numbers and the range is {y|y > 0}.

Holt Algebra 2


Notice as the x-values decrease, the graph of the function gets closer and closer to the x-axis. The function never reaches the x-axis because the value of 2x cannot be zero. In this case, the x-axis is an asymptote. An asymptote is a line that a graphed function approaches as the value of x gets very large or very small.

Holt Algebra 2


A function of the form f(x) = abx, with a > 0 and b > 1, is an exponential growth function, which increases as x increases. When 0 < b < 1, the function is called an exponential decay function, which decreases as x increases.

Holt Algebra 2


5. exponential function6. base7. exponential growth8. exponential decay9. asymptote

Define the Vocabulary(use the slides above or the internet)

Holt Algebra 2


Tell whether the function p(x) = 5(1.2x) shows growth or decay. What is the y-intercept of this function?Step 1 Find the value of the base.

Step 2 The y-intercept is found by plugging a zero in for x.

Notice: It is also the value of “a” so the y-intercept for this function is 5 or (0,5).

Check It Out! Example 1

p(x) = 5(1.2 x) The base , 1.2, is greater than 1. This is an exponential growth function.

p(x) = 5(1.2 0)

p(x) = 5 (1)

p(x) = 5

Holt Algebra 2


Tell whether the function p(x) = 3(.8) x shows growth or decay. What is the y-intercept?

#10

Holt Algebra 2


You can model growth or decay by a constant percent increase or decrease with the following formula:

In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 – r is the decay factor.

Holt Algebra 2


X is used on the graphing calculator for the variable t: Y1 =5000*1.0625^X

Helpful Hint

Holt Algebra 2


In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. What will the population be in the year 2014?

P(t) = a(1 + r)t

Substitute 350 for a and 0.14 for r.

P(t) = 350(1 + 0.14)t

P(t) = 350(1.14)t Simplify.

Exponential growth function.


Holt Algebra 2


Graph the function. Use to find when the population will reach 20,000.

It will take about 31 years for the population to reach 20,000.

Check It Out! Example 2 Continued

Holt Algebra 2


A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100.

f(t) = a(1 – r)t

Substitute 1,000 for a and 0.15 for r.

f(t) = 1000(1 – 0.15)t

f(t) = 1000(0.85)t Simplify.

Exponential decay function.


Holt Algebra 2


Graph the function. Use to find when the value will fall below 100.

It will take about 14.2 years for the value to fall below 100.

Check It Out! Example 3 Continued

200

1000

0

Holt Algebra 2


In 1981, the Australian humpback whale population was 300 and increased at a rate of 12% each year since then. Write a function to model population growth. How many whales will there be after 10 years?

#11

Holt Algebra 2


A motor scooter purchased for $1000 depreciates at an annual rate of 10%. Write an exponential function and graph the function. How much will the scooter be worth after 4 years? Use the graph to predict when the value will fall below $100.

#12

holt algebra 2 7-1 exponential functions, growth, and decay holt algebra 2 read each slide. answer...

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