Page 1
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
ga2nt-04.indd 73 4/16/07 8:57:56 AM
ga2_ntg_04.indd 114ga2_ntg_04.indd 114 4/19/07 3:42:51 PM4/19/07 3:42:51 PM
Page 2
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 Zero exponent property
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
ga2nt-04.indd 74 4/16/07 8:57:57 AM
ga2_ntg_04.indd 115ga2_ntg_04.indd 115 4/19/07 3:42:53 PM4/19/07 3:42:53 PM
Page 3
116 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
5 Zero exponent property
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.
ga2nt-04.indd 75 4/16/07 8:57:58 AM
ga2_ntg_04.indd 116ga2_ntg_04.indd 116 4/19/07 3:42:54 PM4/19/07 3:42:54 PM
Page 4
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 117
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
ga2_ntg_04.indd 117ga2_ntg_04.indd 117 4/19/07 3:42:55 PM4/19/07 3:42:55 PM
Page 5
118 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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 ————————————
ga2_ntg_04.indd 118ga2_ntg_04.indd 118 4/19/07 3:42:56 PM4/19/07 3:42:56 PM
Page 6
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 119
4.2 Perform Function Operations and CompositionGoal p Perform operations with functions.Georgia
PerformanceStandard(s)
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
ga2nt-04.indd 76 4/18/07 3:27:12 PM
ga2_ntg_04.indd 119ga2_ntg_04.indd 119 4/19/07 3:42:57 PM4/19/07 3:42:57 PM
Page 7
120 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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.
ga2nt-04.indd 77 4/16/07 8:58:00 AM
ga2_ntg_04.indd 120ga2_ntg_04.indd 120 4/19/07 3:42:58 PM4/19/07 3:42:58 PM
Page 8
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 121
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.
Checkpoint Complete the following exercise.
ga2nt-04.indd 78 4/16/07 8:58:01 AM
ga2_ntg_04.indd 121ga2_ntg_04.indd 121 4/19/07 3:42:59 PM4/19/07 3:42:59 PM
Page 9
122 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
6. Rework Example 4 for an original price of $800, a $110 coupon, and a 15% discount.
Checkpoint Complete the following exercise.
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
ga2nt-04.indd 79 4/16/07 8:58:02 AM
ga2_ntg_04.indd 122ga2_ntg_04.indd 122 4/19/07 3:43:00 PM4/19/07 3:43:00 PM
Page 10
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 123
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))
ga2_ntg_04.indd 123ga2_ntg_04.indd 123 4/19/07 3:43:01 PM4/19/07 3:43:01 PM
Page 11
124 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
4.2 Practice continued
Name ——————————————————————— Date ————————————
ga2_ntg_04.indd 124ga2_ntg_04.indd 124 4/19/07 3:43:02 PM4/19/07 3:43:02 PM
Page 12
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 125
4.3 Use Inverse FunctionsGoal p Find inverse functions.Georgia
PerformanceStandard(s)
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
ga2nt-04.indd 80 4/18/07 3:27:30 PM
ga2_ntg_04.indd 125ga2_ntg_04.indd 125 4/19/07 3:43:03 PM4/19/07 3:43:03 PM
Page 13
126 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
ga2nt-04.indd 81 4/16/07 8:58:03 AM
ga2_ntg_04.indd 126ga2_ntg_04.indd 126 4/19/07 3:43:04 PM4/19/07 3:43:04 PM
Page 14
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 127
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 .
Checkpoint Complete the following exercise.
3. Find the inverse of f(x) 5 9x2, x ≥ 0.
Checkpoint Complete the following exercise.
ga2nt-04.indd 82 4/16/07 8:58:04 AM
ga2_ntg_04.indd 127ga2_ntg_04.indd 127 4/19/07 3:43:05 PM4/19/07 3:43:05 PM
Page 15
128 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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.
ga2nt-04.indd 83 4/16/07 8:58:05 AM
ga2_ntg_04.indd 128ga2_ntg_04.indd 128 4/19/07 3:43:06 PM4/19/07 3:43:06 PM
Page 16
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 129
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
ga2_ntg_04.indd 129ga2_ntg_04.indd 129 4/19/07 3:43:08 PM4/19/07 3:43:08 PM
Page 17
130 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
4.3 Practice continued
Name ——————————————————————— Date ————————————
ga2_ntg_04.indd 130ga2_ntg_04.indd 130 4/19/07 3:43:09 PM4/19/07 3:43:09 PM
Page 18
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 131
4.4 Graph Exponential Growth FunctionsGoal p Graph and use exponential growth functions.Georgia
PerformanceStandard(s)
MM2A2b, MM2A2c
Your Notes
VOCABULARY
Exponential function
Exponential growth function
Growth factor
Asymptote
End behavior
ga2nt-04.indd 84 4/16/07 8:58:06 AM
ga2_ntg_04.indd 131ga2_ntg_04.indd 131 4/19/07 3:43:10 PM4/19/07 3:43:10 PM
Page 19
132 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
1. Graph y 5 2x and find the average rate of change over the interval 22 ≤ x ≤ 2.
x
y
1
1
Checkpoint Complete the following exercise.
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
ga2nt-04.indd 85 4/16/07 8:58:07 AM
ga2_ntg_04.indd 132ga2_ntg_04.indd 132 4/19/07 3:43:11 PM4/19/07 3:43:11 PM
Page 20
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 133
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.
Checkpoint Complete the following exercise.
Homework
ga2nt-04.indd 86 4/16/07 8:58:08 AM
ga2_ntg_04.indd 133ga2_ntg_04.indd 133 4/19/07 3:43:12 PM4/19/07 3:43:12 PM
Page 21
134 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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 ————————————
ga2_ntg_04.indd 134ga2_ntg_04.indd 134 4/19/07 3:43:14 PM4/19/07 3:43:14 PM
Page 22
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 135
Name ——————————————————————— Date ————————————
LESSON
4.4 Practice continued
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
ga2_ntg_04.indd 135ga2_ntg_04.indd 135 4/19/07 3:43:15 PM4/19/07 3:43:15 PM
Page 23
136 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
4.5 Graph Exponential Decay FunctionsGoal p Graph and use exponential decay functions.Georgia
PerformanceStandard(s)
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
ga2nt-04.indd 87 4/16/07 8:58:09 AM
ga2_ntg_04.indd 136ga2_ntg_04.indd 136 4/19/07 3:43:16 PM4/19/07 3:43:16 PM
Page 24
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 137
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
Checkpoint Complete the following exercise.
ga2nt-04.indd 88 4/16/07 8:58:10 AM
ga2_ntg_04.indd 137ga2_ntg_04.indd 137 4/19/07 3:43:18 PM4/19/07 3:43:18 PM
Page 25
138 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
2. Graph the function. State the domain and range.
x
y
2
2
y 5 3 12}3 2x 1 12 3
Checkpoint Complete the following exercise.
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.
Checkpoint Complete the following exercise.
Homework
ga2nt-04.indd 89 4/16/07 8:58:12 AM
ga2_ntg_04.indd 138ga2_ntg_04.indd 138 4/19/07 3:43:19 PM4/19/07 3:43:19 PM
Page 26
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 139
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)
ga2_ntg_04.indd 139ga2_ntg_04.indd 139 4/19/07 3:43:21 PM4/19/07 3:43:21 PM
Page 27
140 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
4.5 Practice continued
Name ——————————————————————— Date ————————————
ga2_ntg_04.indd 140ga2_ntg_04.indd 140 4/19/07 3:43:22 PM4/19/07 3:43:22 PM
Page 28
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 141
4.6 Solve Exponential Equations and InequalitiesGoal p Solve exponential equations and inequalities.Georgia
PerformanceStandard(s)
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.
Checkpoint Complete the following exercise.
ga2nt-04.indd 90 4/16/07 8:58:13 AM
ga2_ntg_04.indd 141ga2_ntg_04.indd 141 4/19/07 3:43:23 PM4/19/07 3:43:23 PM
Page 29
142 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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.
Checkpoint Complete the following exercise.
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
ga2nt-04.indd 91 4/16/07 8:58:13 AM
ga2_ntg_04.indd 142ga2_ntg_04.indd 142 4/19/07 3:43:24 PM4/19/07 3:43:24 PM
Page 30
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 143
Your Notes
3. Solve 2x 1 1 ≥ 42x 2 1.
Checkpoint Complete the following exercise.
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.
Checkpoint Complete the following exercise.
Homework
ga2nt-04.indd 92 4/16/07 8:58:15 AM
ga2_ntg_04.indd 143ga2_ntg_04.indd 143 4/19/07 3:43:26 PM4/19/07 3:43:26 PM
Page 31
144 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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 ————————————
ga2_ntg_04.indd 144ga2_ntg_04.indd 144 4/19/07 3:43:27 PM4/19/07 3:43:27 PM
Page 32
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 145
Name ——————————————————————— Date ————————————
LESSON
4.6 Practice continued
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.
ga2_ntg_04.indd 145ga2_ntg_04.indd 145 4/19/07 3:43:27 PM4/19/07 3:43:27 PM
Page 33
146 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
PerformanceStandard(s)
MM2A3d
Your Notes
ga2nt-04.indd 93 4/16/07 8:58:15 AM
ga2_ntg_04.indd 146ga2_ntg_04.indd 146 4/19/07 3:43:28 PM4/19/07 3:43:28 PM
Page 34
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 147
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.
ga2nt-04.indd 94 4/16/07 8:58:16 AM
ga2_ntg_04.indd 147ga2_ntg_04.indd 147 4/19/07 3:43:29 PM4/19/07 3:43:29 PM
Page 35
148 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
ga2nt-04.indd 95 4/16/07 8:58:17 AM
ga2_ntg_04.indd 148ga2_ntg_04.indd 148 4/19/07 3:43:30 PM4/19/07 3:43:30 PM
Page 36
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 149
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, . . .
ga2_ntg_04.indd 149ga2_ntg_04.indd 149 4/19/07 3:43:32 PM4/19/07 3:43:32 PM
Page 37
150 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
4.7 Practice continued
Name ——————————————————————— Date ————————————
ga2_ntg_04.indd 150ga2_ntg_04.indd 150 4/19/07 3:43:32 PM4/19/07 3:43:32 PM
Page 38
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 151
4.8 Analyze Arithmetic Sequences and SeriesGoals p Study arithmetic sequences and series.Georgia
PerformanceStandard(s)
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
ga2_ntg_04.indd 151ga2_ntg_04.indd 151 4/19/07 3:43:33 PM4/19/07 3:43:33 PM
Page 39
152 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
ga2nt-04.indd 97 4/16/07 8:58:18 AM
ga2_ntg_04.indd 152ga2_ntg_04.indd 152 4/19/07 3:43:34 PM4/19/07 3:43:34 PM
Page 40
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 153
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
ga2nt-04.indd 98 4/16/07 8:58:19 AM
ga2_ntg_04.indd 153ga2_ntg_04.indd 153 4/19/07 3:43:36 PM4/19/07 3:43:36 PM
Page 41
154 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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 ————————————
ga2_ntg_04.indd 154ga2_ntg_04.indd 154 4/19/07 3:43:37 PM4/19/07 3:43:37 PM
Page 42
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 155
Name ——————————————————————— Date ————————————
LESSON
4.8 Practice continued
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.
ga2_ntg_04.indd 155ga2_ntg_04.indd 155 4/19/07 3:43:38 PM4/19/07 3:43:38 PM
Page 43
156 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
4.9 Analyze Geometric Sequences and SeriesGoal p Study geometric sequences and series.Georgia
PerformanceStandard(s)
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.
Checkpoint Complete the following exercise.
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
ga2nt-04.indd 99 4/16/07 8:58:20 AM
ga2_ntg_04.indd 156ga2_ntg_04.indd 156 4/19/07 3:43:39 PM4/19/07 3:43:39 PM
Page 44
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 157
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.
an 5 a1rn 2 1 Write general rule.
5 a1( ) 2 1 Substitute for an, r, and n.
5 a1 Solve for a1.
So, a rule for the nth term is:
an 5 a1rn 2 1 Write general rule.
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
ga2nt-04.indd 100 4/16/07 8:58:21 AM
ga2_ntg_04.indd 157ga2_ntg_04.indd 157 4/19/07 3:43:40 PM4/19/07 3:43:40 PM
Page 45
158 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
ga2_ntg_04.indd 158ga2_ntg_04.indd 158 4/19/07 3:43:41 PM4/19/07 3:43:41 PM
Page 46
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 159
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
ga2_ntg_04.indd 159ga2_ntg_04.indd 159 4/19/07 3:43:42 PM4/19/07 3:43:42 PM
Page 47
160 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
4.9 Practice continued
Name ——————————————————————— Date ————————————
ga2_ntg_04.indd 160ga2_ntg_04.indd 160 4/19/07 3:43:44 PM4/19/07 3:43:44 PM
Page 48
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 161
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
ga2_ntg_04.indd 161ga2_ntg_04.indd 161 4/19/07 3:43:45 PM4/19/07 3:43:45 PM
Page 49
162 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.
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
ga2_ntg_04.indd 162ga2_ntg_04.indd 162 4/19/07 3:43:46 PM4/19/07 3:43:46 PM