4.1 Use Properties of Exponents · 5 Simplify exponents. 5 Zero exponent property 5 Negative...

114 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.

Use Properties of Exponents4.1Goal p Simplify expressions involving powers.Georgia

PerformanceStandard(s)

MM2A2a

Your Notes

VOCABULARY

Scientific notation

PROPERTIES OF EXPONENTS

Let a and b be real numbers and let m and n be integers.

Product of Powers Property am p an 5 a

Power of a Power Property (am)n 5 a

Power of a Product Property (ab)m 5 a b

Negative Exponent Property a2m 5 , a Þ 0 Zero Exponent Property a0 5 , a Þ 0

Quotient of Powers Property am}an 5 a , a Þ 0

Power of a Quotient Property 1a}b 2m 5 , b Þ 0

1. 185}82 221

Checkpoint Evaluate or simplify the expression.

a. (62)3 5 6 5 6 5

b. 45}43 5 4 5 4 5

c. 724 5 5

Example 1 Evaluate a numerical expression

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Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 115

Your Notes

Iceland Iceland covers about 1.03 3 105 square kilometers and has a population of approximately 2.94 3 105 people. About how many people are there per square kilometer?

Solution

Population}Land area

5 Divide population by land area.

5 Quotient of powers property

ø Use a calculator.

5 Zero exponent property

There are about people per square kilometer.

Example 2 Use scientific notation in real life

a.(x5y2)3}x15y8

5x15y8

Power of a product property

5x15y8

Power of a power property

5 Quotient of powers property

5 Simplify exponents.


5 Negative exponent property

b. 1a24}b2 22 5 Power of a quotient

property

5Power of a power property

5Negative exponent property

Example 3 Simplify expressions

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Your Notes

Beach Ball The radius of a beach ball is about 5.6 times greater than the radius of a baseball. How many times as great as the baseball’s volume is the beach ball’s volume?

Let r represent the radius of the baseball.

Beach ball's volume}}Baseball's volume 5 4

}3πr3

4}3π( )3 The volume of a

sphere is 4}3πr 3.

5 4}3πr3

4}3π Power of a

product property

5 Quotient of powers


ø Approximate power.

The beach ball’s volume is about times as great as the baseball’s volume.

Example 4 Compare real-life volumes

Homework

2. (4 3 103)(7 3 1022)

3. 122a2b}

a5b 23

Checkpoint Simplify or evaluate the expression. Tell which properties of exponents you used.

4. Rework Example 4 where the radius of a beach ball is about 6 times the radius of a baseball.

Checkpoint Complete the following exercise.

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Name ——————————————————————— Date ————————————

LESSON

4.1 PracticeEvaluate the power.

1. 32 2. 53 3. 25 4. 44

5. 90 6. 221 7. 722 8. 1026

Evaluate the expression. Tell which properties of exponents you used.

9. 42 p 43 10. (23)4(23) 11. (52)3

12. (70)5 13. 20 p 225 14. 37

} 34

15. (103)3 16. 1 5 } 6 2

2 17.

(25)6

} 25

18. 82

} 83 19.

92

} 922 20. 1 1 }

2 2

25

Write the number in scientifi c notation.

21. 527,000 22. 0.0000526

23. 0.0023 24. 5,983,000,000,000

25. 17,600,000,000,000,000 26. 0.0000007



Write the answer in scientifi c notation.

27. (3.2 3 104)(1.5 3 105) 28. (5.7 3 1026)(6.2 3 108)

29. (2.8 3 103)2 30. (4.3 3 102)2

31. 8.4 3 1010

} 1.4 3 108 32. 3.6 3 1025

} 4.8 3 1027

Simplify the expression. Tell which properties of exponents you used.

33. b4 p b2 34. x23 p x5

35. (s7)2 36. (5y)2

37. 1 z9

} 3z5 2

21 38.

m2

} m6

39. 1 x }

x2y 2

2 40. 1 n }

4m2n 2

22

41. Earth Science The total volume of water on Earth is about 326,000,000 cubic miles. Write this number in scientifi c notation.

42. National Debt On August 1, 2005, the national debt of the United States was about $7,870,000,000,000. The population of the United States at that time was about 297,000,000. If the national debt was divided evenly among everyone in the country, how much would each person owe? Write your answer in scientifi c notation.

LESSON

4.1 Practice continued

Name ——————————————————————— Date ————————————



4.2 Perform Function Operations and CompositionGoal p Perform operations with functions.Georgia


MM2A5d

Your Notes

VOCABULARY

Power function

Composition

Let f(x) 5 3x2 and g(x) 5 25x2. Find the following.

a. f(x) 1 g(x)

b. f(x) 2 g(x)

c. the domains of f 1 g and f 2 g

Solutiona. f(x) 1 g(x) 5 3x2 1 (25x2)

5

b. f(x) 2 g(x) 5 3x2 2 (25x2)

5

c. The functions f and g each have the same domain: . So, the domains of f 1 g and f 2 g

also consist of .

Example 1 Add and subtract functions

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Your Notes

Let f(x) 5 7x and g(x) 5 x6. Find the following.

a. f(x) p g(x)

b.f(x)}g(x)

c. the domains of f p g and f}g

Solution

a. f(x) p g(x) 5 (7x)(x6) 5

b.f(x)}g(x)

5

c. The domain of f consists of , and the domain of g consists of . So, the domain of f p g consists of .

Because g(0) 5 , the domain of f}g is restricted to

.

Example 2 Multiply and divide functions

1. f(x) 1 g(x)

2. f(x) 2 g(x)

3. f(x) p g(x)

4.f(x)}g(x)

Checkpoint Let f(x) 5 4x3 and g(x) 5 22x3. Perform the indicated operation. State the domain.

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Your Notes

Let f(x) 5 6x21 and g(x) 5 3x 1 5. Find the following.

a. f (g(x)) b. g(f (x)) c. f (f (x))

d. the domain of each composition

Solutiona. f (g(x)) 5 f (3x 1 5) 5

b. g(f(x)) 5 g(6x21)

5

c. f (f (x)) 5 f (6x21) 5

d. The domain of f (g(x)) consists of

except x 5 because g 1 2 5 0 is not in

the . (Note that f(0) 5 , which

is .) The domains of g(f(x)) and f(f(x))consist of except x 5 , again because .

Example 3 Find compositions of functions

5. Let f (x) 5 5x 2 4 and g(x) 5 3x21. Find (a) f (g(x)), (b) g(f (x)), (c) f (f (x)), and (d) the domain of each composition.


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Your Notes

6. Rework Example 4 for an original price of $800, a $110 coupon, and a 15% discount.


Homework

Computer Store You purchase a computer with a price of $700. The computer store applies a store flyer coupon of $100 and a 12% promotional discount. Use composition to find the final price of the purchase when the coupon is applied before the discount. Use composition to find the final price of the purchase when the discount is applied before the coupon.

Step 1 Write functions for the discounts. Let x be the original price, f(x) be the price after the $100 coupon, and g(x) be the price after the 12% promotional discount.

Function for the $100 coupon: f(x) 5

Function for the 12% discount:

g(x) 5 5

Step 2 Compose the functions.

$100 coupon is applied first:

g(f(x)) 5 5

12% discount is applied first:

f(g(x)) 5 5

Step 3 Evaluate the functions g(f(x)) and f(g(x)) when x 5 700.

g(f(700)) 5 5 5

f(g(700)) 5 5 5

The final price is when the $100 coupon is applied before the 12% discount. The final price is when the 12% discount is applied before the $100 coupon.

Example 4 Solve a multi-step problem

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Name ——————————————————————— Date ————————————

LESSON

4.2 PracticeLet f(x) 5 x2 1 2, g(x) 5 3x2 2 1, and h(x) 5 22x2 1 3. Perform the indicated operation. State the domain.

1. f (x) 1 g(x) 2. f (x) 1 h(x)

3. h(x) 1 g(x) 4. f (x) 2 g(x)

5. h(x) 2 f (x) 6. g(x) 2 h(x)

Let f(x) 5 4x3, g(x) 5 2x4, and h(x) 5 26x2. Perform the indicated operation. State the domain.

7. f (x) p g(x) 8. f (x) p h(x)

9. h(x) p g(x) 10. f (x)

} g(x)

11. h(x)

} f (x)

12. h(x)

} g(x)

Let f(x) 5 2x 1 3, g(x) 5 x2 2 1, and h(x) 5 x 1 1 } 5 . Find the indicated value.

13. f (g(1)) 14. h(g(4)) 15. f (h(26))

16. g( f (2)) 17. h( f (23)) 18. g(g(2))



Let f (x) 5 2x21, g(x) 5 2x 1 5, and h(x) 5 x 2 4 } 2 . Perform the

indicated operation.

19. f (g(x)) 20. g(h(x))

21. f (h(x)) 22. g( f (x))

23. h( f (x)) 24. g(g(x))

Let f (x) 5 2x 1 2, g(x) 5 x2, and h(x) 5 3 } x 2 2

. State the domain of

the operation.

25. f (x) 1 g(x) 26. h(x) 2 f (x)

27. h(x) p g(x) 28. g(x)

} f (x)

29. h(g(x)) 30. f (g(x))

31. Profi t A company estimates that its cost C and revenue R can be modeled by the functions C(x) 5 0.6x 1 15,000 and R(x) 5 1.25x where x is the number of units produced. The company’s profi t P is modeled by P(x) 5 R(x) 2 C(x). Find the profi t equation and determine the profi t when 500,000 units are produced.

LESSON


Name ——————————————————————— Date ————————————



4.3 Use Inverse FunctionsGoal p Find inverse functions.Georgia


MM2A5a, MM2A5b, MM2A5c, MM2A5d

Your Notes

VOCABULARY

Inverse relation

Inverse functions

nth root of a

INVERSE FUNCTIONS

Functions f and g are inverse functions of each other provided:

f (g(x)) 5 and g(f (x)) 5

The function g is denoted by f21, read as “f inverse.”

HORIZONTAL LINE TEST

The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f

.

Function Not a function

x

y

x

y

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Your Notes

Verify that f(x) 5 7x 2 4 and f21(x) 51}7

x 14}7 are

inverse functions.

Show that f (f21(x)) 5 x. Show that f21(f (x)) 5 x.

f21(f (x)) 5 f21(7x 2 4)

5

5

5

f (f21(x)) 5 f 11}7x 14}7 2

5

5

5

Example 1 Verify that functions are inverses

1. f (x) 5 23x 1 5

Checkpoint Find the inverse of the function. Then verify that your result and the original function are inverses.

Consider the function f(x) 53}x. Determine whether the

inverse of f is a function. Then find the inverse.

Graph the function. Notice that no

x

y

1

1

horizontal line intersects the graph more than once. The inverse of f is a function. To find an equation for f21, complete the following steps.

y 53}x Replace f(x) with y.

5 2} Switch x and y.

5 3 Multiply each side by y.

5 Divide each side by x.

The inverse of f is f21(x) 5 .

Example 2 Find the inverse of a function of the form y 5 a}x

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Your Notes

Find the inverse of f(x) 5 4x2, x ≤ 0. Then graph fand f21.

f (x) 5 4x2 Write original function.

y 5 4x2 Replace f(x) with y.

Switch x and y.

Divide each side by 4.

Take square roots of each side.

The domain of f is restricted to

x

y

1

1

negative values of x. So, the range of f21 must also be restricted to negative values, and therefore the

inverse is f21(x) 5 . (If the

domain were restricted to x ≥ 0, you

would choose f21(x) 5 .)

Example 3 Find the inverse of a quadratic function

2. Find the inverse of f(x) 54}x .


3. Find the inverse of f(x) 5 9x2, x ≥ 0.


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Your Notes

Homework

Consider the function f(x) 5 5 x7. Determine whether the inverse of f is a function. Then find the inverse.

Solution

x

y

1

1

Graph the function f. Notice that no intersects the graph

more than once. So, the inverse of f is a . To find an equation forf21, complete the following steps.

f(x) 5 5x7 Write original function. y 5 5x7 Replace f(x) with y.

Switch x and y.

Divide each side by .

Take seventh root of each side.

The inverse of f is f21(x) 5 .

Example 4 Find the inverse of a power function

4. f (x) 5 2x4 1 1 5. g(x) 51

}32x5

Checkpoint Find the inverse of the function.

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Name ——————————————————————— Date ————————————

LESSON

4.3 PracticeFind the inverse relation.

1. x 0 1 2 3 4

y 3 5 7 9 11

2. x 0 1 2 3 4

y 2 1 0 21 22

Find an equation for the inverse relation.

3. y 5 x 1 1 4. y 5 5x 5. y 5 2x 2 3

6. y 5 2x 1 6 7. y 5 1 }

2 x 1 4 8. y 5

4 }

3 2

1 }

3 x

Graph the function. Then use the horizontal line test to determine whether the inverse of f is a function.

9. f (x) 5 3x2 1 1 10. f (x) 5 x4 2 2 11. f (x) 5 5 } x

x

y

1

1

x

y

1

1

x

y

2

2



Verify that f and g are inverse functions.

12. f (x) 5 x 1 2; g(x) 5 x 2 2 13. f (x) 5 3x; g(x) 5 1 }

3 x

14. f (x) 5 x3; g(x) 5 3 Ï}

x 15. f (x) 5 4x 2 1; g(x) 5 1 }

4 x 1

1 }

4

16. f (x) 5 1 }

x ; g(x) 5

1 }

x 17. f (x) 5 2x 1

1 }

3 ; g(x) 5

1 }

2 x 2

1 }

6

In Exercises 18 and 19, use the following information.

Conversion The formula to convert miles m to kilometers k is 1.609m 5 k.

18. Write the inverse function, which converts kilometers to miles.

19. How many miles is 40 kilometers? Round your answer to two decimal places.

In Exercises 20 and 21, use the following information.

Geometry The formula C 5 2πr gives the circumference of a circle of radius r.

20. Write the inverse function, which gives the radius of a circle of circumference C.

21. What is the radius of a circle with a circumference of 14 inches? Round your answer to two decimal places.

LESSON


Name ——————————————————————— Date ————————————



4.4 Graph Exponential Growth FunctionsGoal p Graph and use exponential growth functions.Georgia


MM2A2b, MM2A2c

Your Notes

VOCABULARY

Exponential function

Exponential growth function

Growth factor

Asymptote

End behavior

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Your Notes

1. Graph y 5 2x and find the average rate of change over the interval 22 ≤ x ≤ 2.

x

y

1

1


Graph y 5 3x. Analyze the graph. Find the average rate of change over the interval 22 ≤ x ≤ 2.

Step 1 Make a table of values.

x

y

2

1

x 22 21 0 1 2

y

Step 2 Plot the points from the table.

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

Step 4 Examine the graph. The graph intersects the y-axis at point . So the y-intercept of y 5 3x is . You can see from the graph that as the value of x approaches 2` the value of the function approaches but never reaches . So the function has no zeros or x-intercepts. As the value of x approaches 1`, the value of the function approaches . Also, the function is increasing on the interval . Using the points

122, 1}9 2, and (2, 9), the average rate of change

over the interval 22 ≤ x ≤ 2 is , or about .

Example 1 Graph y 5 bx for b > 1

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Your Notes

Graph y 5 2 p 3x 2 2 2 2. State the domain and range.

SolutionBegin by sketching the graph of

x

y

1

1

y 5 2 p 3x, which passes through (0, ) and (1, ). Then translate the graph and

.

The graph’s asymptote is the line . The domain is all real

, and the range is .

Example 3 Graph y 5 abx 2 h 1 k for b > 1

Graph the function y 51}4p 6x.

Solution

x

y

1

1

Plot 10, 2 and 11, 2. Then, from

left to right, draw a curve that begins just the x-axis, passes through the two points, and moves

.

Example 2 Graph y 5 abx for b > 1

2. Graph the function

x

y

1

1

y 5 2 p 4x 1 1 2 3. State the domain and range.


Homework

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Match the function with its graph.

1. f (x) 5 2x 2. f (x) 5 22x 3. f (x) 5 4(2x)

4. f (x) 5 1 }

2 (2x) 5. f (x) 5 2

1 } 2 (2x) 6. f (x) 5 24(2x)

A.

x

y

2

2

(1, 22)(0, 21)

B.

x

y

2

2(0, 1)(1, 2)

C.

x

y

2

4

(1, 21)

( )120, 2

D.

x

y

4

2

(0, 4)

(1, 8)

E.

x

y

2

2

(0, 24)

(1, 28)

F.

x

y

2

2

(1, 1)( )120,

LESSON

4.4 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Graph the function.

7. f (x) 5 4x 8. f (x) 5 3 p 3x 9. f (x) 5 25x

x

y

2

2

x

y

2

2

x

y

2

2

Graph the function. State the domain and range. Describe the end behavior. Find the rate of change over the interval 21 ≤ x ≤ 1.

10. f (x) 5 2x 1 2 11. f (x) 5 3x 1 1 12. f (x) 5 2x 2 2 2 1

x

y

2

2

x

y

2

2

x

y

2

2



4.5 Graph Exponential Decay FunctionsGoal p Graph and use exponential decay functions.Georgia


MM2A2b, MM2A2c, MM2A2e

Your Notes

VOCABULARY

Exponential decay function

Decay factor

Graph y 5 11}2 2x. Analyze the graph. Find the average rate

of change over the interval 22 ≤ x ≤ 2.

Step 1 Make a table of values.

x

y

1

1

x 22 21 0 1 2

y

Step 2 Plot the points from the table.

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

Step 4 Examine the graph. The y-intercept of the function is . As the value of x approaches 2` the value of the function approaches . As the value of x approaches 1`, the value of the function approaches . Also, the function is decreasing on the interval . Using

the points (22, 4), and 12, 1}4 2, the average rate of

change over the interval 22 ≤ x ≤ 2 is , or

about .

Example 1 Graph y 5 bx for 0 < b < 1

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Your Notes

Graph the function y 5 2213}4 2x.Plot (0, ) and 11, 2. Then,

x

y1

1

from right to left, draw a curve that begins just the x-axis, passes through the two points, and moves to the left.

Example 2 Graph y 5 abx for 0 < b < 1

Graph y 5 2 13}5 2x 2 11 1. State the domain and range.

SolutionBegin by sketching the graph of

x

y

1

1

y 5 2 13}5 2x, which passes through (0, )

and 11, 2. Then translate the graph

and . Notice that the graph passes through (1, )

and 12, 2.The graph’s asymptote is the line . The domain is , and the range is .

Example 3 Graph y 5 abx 2 h 1 k for 0 < b < 1

1. Graph y 5 11}4 2x and find the average rate of change

over the interval 22 ≤ x ≤ 2.

x

y

4

1


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Your Notes

2. Graph the function. State the domain and range.

x

y

2

2

y 5 3 12}3 2x 1 12 3


Televisions A new television costs $1200. The value of the television decreases by 21% each year. Write an exponential decay model giving the television’s value y(in dollars) after t years. Estimate the value after 2 years.

Solution

The initial amount is a 5 and the percent decrease is r 5 . So, the model is:

y 5 a(1 2 r)t Write exponential decay formula.5 Substitute for a and r.5 Simplify.

When t 5 2, the television’s value is y 5 1200(0.79)2 5 .

Example 4 Find a value after depreciation

3. Rework Example 4, with a 12% decrease each year.


Homework

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Name ——————————————————————— Date ————————————

LESSON

4.5 PracticeTell whether the function represents exponential growth or exponential decay.

1. f (x) 5 1 3 } 4 2

x 2. f (x) 5 1 5 }

3 2

x 3. f (x) 5 5x

4. f (x) 5 1 }

2 (3x) 5. f (x) 5 0.8x 6. f (x) 5

1 }

3 1 3 }

2 2

x

Match the function with its graph.

7. f (x) 5 1 1 } 2 2

x 8. f (x) 5 2 1 1 } 2 2

x 9. f (x) 5 3 1 1 }

2 2

x

10. f (x) 5 1 }

3 1 1 }

2 2

x 11. f (x) 5 23 1 1 }

2 2

x 12. f (x) 5 2

1 } 3 1 1 }

2 2

x

A.

x

y1

1

( )321, 2

(0, 23)

B.

x

y

2

1

( )121,

(0, 1)

C.

x

y

211

( )161,

( )130,

D.

x

y

1

22 ( )161, 2

( )130, 2

E.

x

y

1

1

( )321,

(0, 3)

F.

x

y

1

21

( )121, 2 (0, 21)



Graph the function. State the domain and range. Describe the end behavior of the graph. Find the rate of change over 21 ≤ x ≤ 1.

13. f (x) 5 1 2 } 3 2

x 14. f (x) 5 2 1 2 } 3 2

x 1 2

x

y

2

2

x

y

2

2

15. f (x) 5 1 1 } 4 2

x 2 3 16. f (x) 5 1 1 } 2 2 x 1 2

1 1

x

y

2

2

x

y

1

1

17. Depreciation You buy a new computer and accessories for $1200. The value of the computer decreases by 30% each year. Write an exponential decay model giving the computer’s value V (in dollars) after t years. What is the value of the computer after four years?

LESSON


Name ——————————————————————— Date ————————————



4.6 Solve Exponential Equations and InequalitiesGoal p Solve exponential equations and inequalities.Georgia


MM2A2d

Your Notes

VOCABULARY

Exponential equation

Exponential inequalities in one variable

Solve 64x 5 16x 1 1.

64x 5 16x 1 1 Write original equation.

( )x 5 ( )x 1 1 Rewrite each power with base .

5 Power of a power property

5 Property of equality

x 5 Solve for x.

The solution is .

CHECK Substitute the solution into the original equation.

64 0 16 1 1 Substitute for x.

5 Solution checks.

Example 1 Solve by equating exponents

1. Solve 37x 2 3 5 92x.


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Your Notes

Solve 2x 5 11}4 2x 2 3.

Graph y 5 2x and y 5 11}4 2x 2 3

x

y

1

1

in the same coordinate plane. The graphs intersect only once, when x 5 . So, is the only solution.

Example 2 Solve an exponential equation by graphing

2. Solve 4x 5 11}2 2x 2 1.


Solve 34z ≤ 9z 1 3.

34z ≤ 9z 1 3 Write original inequality.

( )4z ≤ ( )z 1 3 Rewrite each power with base .

≤ Power of a power property

≤ Because f(x) 5 3x is an increasing function, f(x1) ≤ f(x2) implies that x1 ≤ x2.

z ≤ Solve for z.

CHECK Check that the solution is reasonable by substituting several values into the original inequality.

Subsitute z 5 2. Subsitute z 5 4.

34(2) ≤? 92 1 3 34(4) ≤? 94 1 3

Because z 5 is a solution of the inequality and z 5 is not, the soution z ≤ is reasonable.

Example 3 Solve an exponential inequality

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Your Notes

3. Solve 2x 1 1 ≥ 42x 2 1.


Solve 32x 2 1 2 3 > 0.75.

Graph y 5 32x 2 1 2 3 and y 5 0.75

x

y

1

1

in the same coordinate plane. The graphs intersect only once, when x 5 . The graph of y 5 32x 2 1 2 3 is the graph of y 5 0.75 when .

The solution is .

Example 4 Solve an exponential inequality by graphing

4. Solve 2x 2 1 2 4 < 0.5.


Homework

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Solve the exponential equation.

1. 6x 5 63x 2 4 2. 5x 1 3 5 5x 1 3 3. 33x 5 32x 1 1

4. 2x 1 3 5 4x 2 2 5. 34x 5 9x 1 1 6. 48x 2 4 5 16x 1 1

Solve the exponential inequality.

7. 3x 1 3 ≤ 32x 2 4 8. 43x ≤ 4x 1 2 9. 52x 1 1 ≤ 53x 2 3

10. 43x 2 3 ≥ 2x 1 4 11. 5x 1 2 ≥ 25x 2 2 12. 39x ≤ 94x 1 3

LESSON

4.6 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Solve the exponential equation using a graph. Round your answer to the nearest hundredth.

13. 22x 1 3 5 4x 1 1 1 16 14. 34x 2 10 5 7x 2 2 1 2 15. 42x 2 1 5 93x 2 4 2 17

Solve the exponential inequality using a graph. Round your answer to the nearest hundredth.

16. 3x 1 2 2 2 ≤ 52x 17. 1 }

2 2x 2 3

1 1 ≥ 3 18. 3x 1 10 ≤ 4x 1 3

19. Virus An infectious virus is defi ned by its infectivity, or how contagious the virus is to humans. The number of people (in thousands) expected to contract the virus within 6 months is modeled by y 5 1.04(8.35)x where x is the infectivity rating of the virus. How low must the infectivity rating be so no more than 200,000 people become infected within 6 months? How high must the infectivity rating be so more than 700,000 become infected within 6 months? Round your answers to the nearest hundredth.



VOCABULARY

Sequence

Terms

Series

Summation notation

Sigma notation

SEQUENCES

A sequence is a function whose domain is a set of integers. If a domain is not specified, it is understood that the domain starts with 1. The values in the range are called the of the sequence.

Domain: 1 2 3 4 . . . n The relative position of each term

Range: a1 a2 a3 a4 . . . an Terms of the sequence

A sequence has a limited number of terms. An sequence continues without stopping.

Finite sequence: 2, 4, 6, 8 Infinite Sequence: 2, 4, 6, 8, . . .

A sequence can be specified by an equation, or .For example, both sequences above can be described by the rule an 5 2n or f(n) 5 2n.

4.7 Define and Use Sequences and SeriesGoal p Recognize and write rules for number patterns.Georgia


MM2A3d

Your Notes

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Your Notes

Write the first six terms of an 5 2n 1 1.

a1 5 5 1st term

a2 5 5 2nd term

a3 5 5 3rd term

a4 5 5 4th term

a5 5 5 5th term

a6 5 5 6th term

Example 1 Write terms of sequences

Describe the pattern, write the next term, and write a rule for the nth term of the sequence (a) 1, 4, 9, 16, . . . and (b) 0, 7, 26, 63, . . ..

Solution

a. You can write the terms as 2, 2, 2, 2, . . .. The next term is a5 5 5 . A rule for the nthterm is an 5 .

b. You can write the terms as 2 1, 2 1, 2 1, 2 1, . . .. The next term is

a5 5 2 1 5 . A rule for the nth term is an 5 .

Example 2 Write rules for sequences

1. Write the first six terms of the sequence f (n) 5 3n 2 7.

2. For the sequence 23, 9, 227, 81, . . ., describe the pattern, write the next term, and write a rule for the nth term.

Checkpoint Complete the following exercises.

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Your Notes

Write the series using summation notation.

a. 4 1 7 1 10 1 . . . 1 46 b. 1 1 1}8 1

1}27 1

1}64 1 . . .

Solutiona. Notice that the first term is 3(1) 1 1, the second

is , the third is , and the last is . So, ai 5 where i 5 1, 2, 3, . . ., . The lower limit of summation is and the upper limit of summation is .

The summation notation for the series is .

b. Notice that for each term, the denominator is a perfect

cube. So, ai 5 where i 5 1, 2, 3, 4, . . .. The lower

limit of summation is and the upper limit of summation is .

The summation notation for the series is .

Example 3 Write series using summation notation

3. 7 1 14 1 21 1 . . . 1 77

4. 24 2 8 2 12 2 16 2 . . .

Checkpoint Write the series using summation notation.

SERIES

In the seriesi 5 1∑ 4

2i, i is the ,

1 is the , and 4 is the .

Homework

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Name ——————————————————————— Date ————————————

LESSON

4.7 PracticeWrite the fi rst six terms of the sequence.

1. an 5 n 1 1 2. an 5 3n 3. an 5 23n

4. f (n) 5 n2 1 3 5. f (n) 5 10

} n 6. f (n) 5

n }

n 1 2

For the sequence, describe the pattern, write the next term, and write a rule for the nth term.

7. 4, 8, 12, 16, . . . 8. 7, 11, 15, 19, . . .

9. 23, 6, 29, 12, . . . 10. 3, 9, 27, 81, . . .

11. 1 }

11 ,

2 }

11 ,

3 }

11 ,

4 }

11 , . . . 12.

2 }

3 ,

2 }

9 ,

2 }

27 ,

2 }

81 , . . .

13. 20.9, 0.2, 1.3, 2.4, . . . 14. 2.1, 4.2, 8.4, 16.8, . . .



Graph the sequence.

15. 26, 24, 22, 0, 2 16. 0.1, 0.2, 0.4, 0.8, 1.6 17. 9, 3, 1, 1 }

3 ,

1 }

9

n

2

2

an

n

0.3

1

an

n

2

1

an

Write the series using summation notation.

18. 10 1 20 1 30 1 40 1 50 19. 1 2 4 1 9 2 16 1 . . . 20. 15 1 1 1 (213) 1 (227)

Find the sum of the series.

21. n 5 1

∑ 5

n2 22. i 5 2

∑ 7

3i 2 5 23. k 5 1

∑ 5

k } 5

24. Amoebas A Petri dish contains three amoebas. An amoeba is a microorganism that reproduces using the process of fi ssion, by simply dividing itself into two smaller amoebas. Once the new amoebas mature, they will go through the same process.

a. Write the terms of the sequence describing the fi rst four generations of the amoeba in the Petri dish.

b. Write a rule for the nth term of the sequence.

c. Find the 10th term of the sequence and describe in words what this term represents.

LESSON


Name ——————————————————————— Date ————————————



4.8 Analyze Arithmetic Sequences and SeriesGoals p Study arithmetic sequences and series.Georgia


MM2A3d, MM2A3e

Your Notes

VOCABULARY

Arithmetic sequence

Common difference

Arithmetic series

Tell whether the sequence 25, 23, 21, 1, 3, . . . is arithmetic.

Find the differences of consecutive terms.

a2 2 a1 5 5

a3 2 a2 5 5

a4 2 a3 5 5

a5 2 a4 5 5

Each difference is , so the sequence arithmetic.

Example 1 Identify arithmetic sequences

RULE FOR AN ARITHMETIC SEQUENCE

The nth term of an arithmetic sequence with first term a1 and common difference d is given by:

an 5 a1 1 (n 2 1)d

THE SUM OF A FINITE ARITHMETIC SERIES

The sum of the first n terms of an arithmetic series is:

Sn 5 n1a1 1 an}

2 2In words, Sn is the of the terms, by .

ga2nt-04.indd 96 4/16/07 8:58:18 AM



Your Notes

1. 32, 27, 21, 17, 10, . . .

Checkpoint Tell whether the sequence is arithmetic.

Write a rule for the nth term of the sequence 2, 9, 16, 23, . . .. Then find a19.

SolutionThe sequence is arithmetic with first term a1 5 2 and common difference d 5 5 . So, a rule for the nth term is:

an 5 a1 1 (n 2 1)d Write general rule.

5 1 (n 2 1) Substitute for a1 and d.

5 Simplify.

The 19th term is a19 5 5 .

Example 2 Write a rule for the nth term

Find the sum of the first 30 terms of the arithmetic series 2 1 4 1 6 1 8 1 . . ..

The series is arithmetic with first term a1 5 2 and common difference d 5 5 . So, a rule for the nth term is:

an 5 a1 1 (n 2 1)d Write general rule.

5 1 (n 2 1) Substitute for a1 and for d.

5 Simplify.

The 30th term is a30 5 5 . The sum of the first 30 terms is:

S30 5 30 1a1 1 a30}

2 2 Write rule for S30.

5 30 Substitute for a1 and for a30.

5 Simplify.

The sum of the first 30 terms is .

Example 3 Find the sum of an arithmetic series

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Your Notes

2. Write a rule for the nth term of the sequence9, 5, 1, 23, . . .. The find a22.

3. Find the sum of the first 20 terms of the arithmetic series 24 1 0 1 4 1 8 1 . . ..

Checkpoint Complete the following exercises.

Find a formula for the sum of the series 4 1 6 1 8 1 . . . 1 (2n 2 2) 1 2n 1 (2n 1 2).

The sequence is arithmetic with first term a1 5 2 and common difference d 5 5 . So, a rule for the nth term is:

an 5 a1 1 (n 2 1)d 5 1 (n 2 1) 5

The sum of the first n terms is:

Sn 5 n1a1 1 an}

2 2 Write rule for Sn.

5 Substitute for a1 and for an.

Sn represents the partial sum of the first n terms of this series. Note that Sn is a quadratic function that can be written as Sn 5 .

Example 4 Find a formula for the sum of a series

4. Find a formula for the partial sum of the series 5 1 10 1 15 1 20 1 . . . 1 (5n 2 5) 1 5n.

Checkpoint Complete the following exercise.Homework

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Tell whether the sequence is arithmetic. Explain why or why not.

1. 1, 4, 7, 10, 13, . . . 2. 1, 3, 7, 15, 31, . . . 3. 23, 21, 1, 3, 5, . . .

4. 8, 4, 0, 24, 28, . . . 5. 1, 22, 3, 24, 5, . . . 6. 1 }

2 ,

1 }

3 ,

1 }

4 ,

1 } 5 ,

1 }

6 , . . .

Write a rule for the nth term of the arithmetic sequence. Then fi nd a8.

7. 7, 10, 13, 16, . . . 8. 8, 9, 10, 11, 12, . . . 9. 3, 1, 21, 23, 25, . . .

10. d 5 2, a5 5 1 11. d 5 4, a4 5 16 12. d 5 25, a10 5 260

Write a rule for the nth term of the arithmetic sequence that has the two given terms.

13. a8 5 21, a10 5 27 14. a3 5 12, a9 5 18 15. a5 5 15, a8 5 30

16. a2 5 25, a7 5 50 17. a12 5 0, a19 5 28 18. a10 5 150, a20 5 100

LESSON

4.8 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Find the sum of the arithmetic series.

19. i 5 1

∑ 4

(i 1 1) 20. i 5 1

∑ 6

3i 21. i 5 1

∑ 12

(2i 1 1)

22. i 5 1

∑ 7

(3i 2 4) 23. i 5 1

∑ 8

2i 24. i 5 16

∑ 20

(5 2 i)

Write a rule for nth term of the sequence whose graph is shown.

25.

n

an

2

1

(1, 7)(2, 5)

(3, 3)(4, 1)

(5,−1)

26.

n

an

2

1(1, −2)

(2, 0)(3, 2)

(4, 4)(5, 6)

27.

n

an

1

1

(1, 5)(2, 4)

(3, 3)(4, 2)

(5, 1)

28. Weightlifting You are trying to fi nd the maximum weight that you can lift in a weightlifting exercise. You start with a single lift of 125 pounds. Then you increase the weight by 2 pounds and try again. You repeat this procedure until you reach a weight that you are unable to lift.

a. Write a rule for the total weight of your nth lifting attempt.

b. You are unable to lift the weight on your sixth lift. So, based on your fi fth lift, what is the maximum amount of weight that you can lift in this exercise?

c. Find the sum of the weights lifted in your fi ve successful lifts.



4.9 Analyze Geometric Sequences and SeriesGoal p Study geometric sequences and series.Georgia


MM2A2f

Your Notes

VOCABULARY

Geometric sequence

Common ratio

Geometric series

Tell whether the sequence 1, 24, 16, 264, 256, . . . is geometric.

Find the ratios of consecutive terms. If the ratios are constant then the sequence is geometric.a2}a1

524}1 5 24

a3}a2

5 5

a4}a3

5 5a5}a4

5 5

Each ratio is , so the sequence geometric.

Example 1 Identify geometric sequences

1. Tell whether the sequence 512, 128, 64, 8, . . . is geometric.


RULE FOR A GEOMETRIC SEQUENCE

The nth term of a geometric sequence with first term a1 and common ratio r is given by: an 5 a1r n 2 1

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Your Notes

Write a rule for the nth term of the sequence 972, 2324, 108, 236, . . .. Then find a10.

Solution

The sequence is geometric with first term a1 5

and common ratio r 5 5 . So, a rule for the

nth term is:

an 5 a1rn 2 1 Write general rule.

5 1 2n 2 1Substitute for a1 and r.

The 10th term is a10 5 5 .

Example 2 Write a rule for the nth term

One term of a geometric sequence is a3 5 218.The common ratio is r 5 3. (a) Write a rule for the nth term. (b) Graph the sequence.

a. Use the general rule to find the first term.


5 a1( ) 2 1 Substitute for an, r, and n.

5 a1 Solve for a1.

So, a rule for the nth term is:


5 Substitute for a1 and r.

b. Create a table of values for the n

an

2201sequence. Notice that the points

lie on an exponential curve.

n 1 2 3

an

n 4 5

an

Example 3 Write a rule given a term and common ratio

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Your Notes

2. 14, 28, 56, 112, . . .

3. a5 5 324, r 5 23

Checkpoint Write a rule for the nth term of the geometric sequence. Then find a9.

THE SUM OF A FINITE GEOMETRIC SERIES

The sum of the first n terms of a finite geometric series with common ratio r Þ 1 is:

Sn 5 a111 2 rn}1 2 r 2

Find the sum of the geometric series i 5 1∑ 13

3(4)i 2 1.

a1 5 5 Identify first term.

r 5 Identify common ratio.

S13 5 a111 2 r13}1 2 r 2 Write rule for S13.

5 5 Substitute and simplify.

Example 4 Find the sum of a geometric series

4. Find the sum of the geometric series i 5 1∑ 11

7(25)n 2 1.

Checkpoint Complete the following exercise.Homework

ga2nt-04.indd 101 4/16/07 8:58:22 AM



Name ——————————————————————— Date ————————————

LESSON

4.9 PracticeTell whether the sequence is geometric. Explain why or why not.

1. 2, 4, 8, 16, 32, . . . 2. 1, 10, 100, 1000, 10,000, . . . 3. 1, 3, 9, 27, 81, . . .

4. 16, 8, 2, 0.5, 0.125, . . . 5. 25, 5, 1, 1 } 5 ,

1 }

25 , . . . 6.

3 }

4 ,

1 }

4 ,

1 }

12 ,

1 }

36 ,

1 }

108 , . . .

Write a rule for the nth term of the geometric sequence. Then fi nd a6.

7. 4, 8, 16, 32, . . . 8. 5000, 500, 50, 5, . . . 9. 3, 12, 48, 192, . . .

Write a rule for the nth term of the geometric sequence. Then graph the fi rst fi ve terms of the sequence.

10. r 5 2, a1 5 1 11. r 5 3, a2 5 15 12. r 5 2 } 3 , a2 5 54

n

3

1

an

n

100

1

an

n

20

1

an



Write a rule for the nth term of the geometric sequence that has the two given terms.

13. a1 5 4, a2 5 12 14. a2 5 2, a5 5 16 15. a3 5 232, a6 5 22048

16. a3 5 1, a6 5 1 } 27 17. a2 5 10, a5 5 80 18. a2 5 20, a4 5

5 }

4

Find the sum of the geometric series.

19. i 5 1

∑ 6

2(2)i 2 1 20. i 5 1

∑ 5

1(3)i 2 1 21. i 5 1

∑ 8

0.5(2)i 2 1

22. i 5 1

∑ 5

1 }

1000 (10)i 2 1 23.

i 5 1 ∑

5

400 1 1 } 2 2

i 2 1 24.

i 5 1 ∑

6

1000 1 4 } 5 2 i 2 1

25. Production A company plans to increase production of a product by 10% each year over the next 12 years. The company will produce 70,000 units next year (year 1).

a. Write a rule giving the number of units produced by the company in year n.

b. Find the numbers of units produced in years 4, 8, and 12.

c. Find the total number of units produced over the next 12 years.

LESSON


Name ——————————————————————— Date ————————————



Words to ReviewGive an example of the vocabulary word.

Scientific notation

Composition

Inverse function

Exponential function

Growth factor

End behavior

Power function

Inverse relation

nth root of a

Exponential growth function

Asymptote

Exponential decay function

ga2nt-04.indd 102 4/16/07 8:58:23 AM



Decay factor

Exponential inequality in one variable

Terms

Summation notation

Arithmetic sequence

Arithmetic series

Common ratio

Exponential equation

Sequence

Series

Sigma notation

Common difference

Geometric sequence

Geometric series

ga2nt-04.indd 103 4/16/07 8:58:24 AM


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