algebra ii 4-1 study guide of...

Algebra II 4-1 Study Guide Page ! of !1 7

Exponential Functions, Growth and Decay Attendance Problems. Evaluate the following. Round appropriately.

1. ! 2. ! 3. ! !!

4. ! !!I can write and evaluate exponential expressions to model growth and decay situations. !

!Common Core • CCSS.MATH.CONTENT.HSF.IF.C.7.E Graph exponential and logarithmic functions,

showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

• CCSS.MATH.CONTENT.HSF.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

• CCSS.MATH.CONTENT.HSA.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.* !

Moore’s law, a rule used in the computer industry, states that the number of transistors per integrated circuit (the processing power) doubles every year. Beginning in the early days of integrated circuits, the growth in capacity may be approximated by this table.

100 1.08( )20 100 0.95( )25 100 1− 0.02( )10

100 1+ 0.08( )−10

Vocabulary

Exponential Function Base

Asymptote Exponential Growth

Exponential Decay


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.

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

In the function y = bx, y is a function of x because the value of y depends on the value of x.

Remember!

Negative exponents indicate a reciprocal. For example:

Remember!


Example 1. Tell whether the function shows growth or decay. Then graph.

A. !

!!!

f x( ) = 10 34

⎛⎝⎜

⎞⎠⎟x

42-4 -2

2

-2

4

0

y

x

30

25

-10

-5

1E X A M P L E Graphing Exponential Functions

Tell whether the function shows growth or decay. Then graph.

A f (x) = 1. 5 x

Step 1 Find the value of the base.f (x) = 1. 5 x The base, 1.5, is greater than 1. This is

an exponential growth function.

Step 2 Graph the function by using a table of values.

x -2 -1 0 1 2 3 4

f (x) 0.4 0.7 1 1.5 2.3 3.4 5.1

B g (x) = 30 (0. 8 x )

Step 1 Find the value of the base.g (x) = 30 (0. 8 x ) The base, 0.8, is less

than 1. This is an exponential decay function.

Step 2 Graph the function by using a graphing calculator.

1. Tell whether the function p (x) = 5 (1. 2 x ) shows growth or decay. Then graph.

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

Initial amount Number of time periods

Final amount Rate of increase

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

When a function increases by a constant rate, such as 7%, this is the same as multiplying by 100% + 7%, or 107% .

In decimal form, I would multiply by 1 + 0.07, or 1.07.

Growth and Decay

Angela Jones,Independence High School

When a function decreases by a constant rate, such as 12%, this is the same as multiplying by 100% – 12%, or 88% .

In decimal form, it’s (1 - 0.12) , or 0.88.

Negative exponents indicate a reciprocal. For example:

x -2 = 1 _ x 2

© P

unch

Sto

ck

© P

unch

Sto

ck

8

12

16

0 8 4 -8 -4

4- 1 Exponential Functions, Growth, and Decay 235

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B. ! !!5. Guided Practice. Tell whether the function p(x) = 5(1.2x) shows growth or decay. Then graph. ! !!!!!!!!!!!!!!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.

g x( ) = 100 1.05( )x


Example 2. Clara invests $5000 in an account that pays 6.25% interest per year. After how many years will her investment be worth $10,000? !!!!!

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

Helpful Hint

0 15

0

100,000

0 20

0

30,000

2E X A M P L E Economics Application

Tony purchased a rare 1959 Gibson Les Paul guitar in 2000 for $12,000. Experts estimate that its value will increase by 14% per year. Use a graph to find when the value of the guitar will be $60,000.

Step 1 Write a function to model the growth in value for this guitar.

f (t) = a (1 + r) t Exponential growth function

= 12,000 (1 + 0.14) t Substitute 12,000 for a and 0.14 for r.

= 12,000 (1.14) t

Step 2 Graph the function.

When graphing exponential functions in an appropriate domain, you may need to adjust the range a few times to show the key points.

Step 3 Use the graph to predict when the value of the guitar will reach $60,000.

Use the feature to find the t-value where f (t) ≈ 60,000.

The function value is approximately 60,000 when t ≈ 12.29. The guitar will be worth $60,000 about 12.29 years after it is purchased, or sometime in 2012.

2. In 1981, the Australian humpback whale population was 350 and has increased at a rate of about 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000.

3E X A M P L E Depreciation Application

The value of a truck bought new for $28,000 decreases 9.5% each year. Write an exponential function, and graph the function. Use the graph to predict when the value will fall to $5000.

Write a function to model the growth in value for this truck.

f (x) = a (1 - r) t Exponential decay function

= 28,000 (1 - 0.095 ) t Substitute 28,000 for a and 0.095 for r.

= 28,000 (0.905) t Simplify.

Graph the function. Use to find when the value of the truck will fall below $5000.

It will take about 17.3 years for the value to drop to $5000.

3. 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.

X is used on the graphing calculator for the variable t:Y1=12000*1.14^X

236 Chapter 4 Exponential and Logarithmic Functions

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6. Guided Practice. 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. Use a graph to predict when the population will reach 20,000. !!

Example 3. A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function and graph the function. Use the graph to predict when the population will drop below 8000. !!!!!!

0 15

0

100,000

0 20

0

30,000

2E X A M P L E Economics Application

Tony purchased a rare 1959 Gibson Les Paul guitar in 2000 for $12,000. Experts estimate that its value will increase by 14% per year. Use a graph to find when the value of the guitar will be $60,000.

Step 1 Write a function to model the growth in value for this guitar.

f (t) = a (1 + r) t Exponential growth function

= 12,000 (1 + 0.14) t Substitute 12,000 for a and 0.14 for r.

= 12,000 (1.14) t

Step 2 Graph the function.

When graphing exponential functions in an appropriate domain, you may need to adjust the range a few times to show the key points.

Step 3 Use the graph to predict when the value of the guitar will reach $60,000.

Use the feature to find the t-value where f (t) ≈ 60,000.

The function value is approximately 60,000 when t ≈ 12.29. The guitar will be worth $60,000 about 12.29 years after it is purchased, or sometime in 2012.

2. In 1981, the Australian humpback whale population was 350 and has increased at a rate of about 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000.

3E X A M P L E Depreciation Application

The value of a truck bought new for $28,000 decreases 9.5% each year. Write an exponential function, and graph the function. Use the graph to predict when the value will fall to $5000.

Write a function to model the growth in value for this truck.

f (x) = a (1 - r) t Exponential decay function

= 28,000 (1 - 0.095 ) t Substitute 28,000 for a and 0.095 for r.

= 28,000 (0.905) t Simplify.

Graph the function. Use to find when the value of the truck will fall below $5000.

It will take about 17.3 years for the value to drop to $5000.

3. 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.

X is used on the graphing calculator for the variable t:Y1=12000*1.14^X

236 Chapter 4 Exponential and Logarithmic Functions

CS10_A2_MESE612263_C04L01.indd Sec1:236CS10_A2_MESE612263_C04L01.indd Sec1:236 2/24/11 3:01:04 PM2/24/11 3:01:04 PM


7. Guided Practice. 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. !!!!!!!!!4-1 Assignment • (p 238) 7-15, 31-34 • (p 241) 1-4.

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