392 Chapter 5 Quadratic Functions Complete the sentences below with vocabulary words from the list above. 1. The number 5i can be classified as both a(n) −−−−−− ? and a −−−−−− ? . 2. The value of the input x that makes the output f (x) equal zero is called the −−−−−− ? . 3. The −−−−−− ? is the point at which the parabola intersects the axis of symmetry. 4. The type and number of solutions to a quadratic equation can be determined by finding the −−−−−− ? . 5. When a parabola opens upward, the y-value of the vertex is the −−−−−− ? of a quadratic function. Graph each function by using a table. 6. f (x) = - x 2 - 2x 7. f (x) = 1 _ 2 x 2 + 3x - 4 Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function. 8. g (x) = 4 (x - 2) 2 9. g (x) = -2 (x + 1) 2 10. g (x) = 1 _ 3 x 2 - 3 11. g (x) = - (x + 2) 2 + 6 Use the description to write each quadratic function in vertex form. 12. f (x) = x 2 is reflected across the x-axis and translated 3 units down to create g. 13. f (x) = x 2 is vertically stretched by a factor of 2 and translated 4 units right to create g. 14. f (x) = x 2 is vertically compressed by a factor of 1 __ 4 and translated 1 unit left to create g. ■ Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph g (x) = 1 __ 2 x 2 + 3. g (x) = 1 __ 2 x 2 + 3 is f vertically compressed by a factor of 1 __ 2 and translated 3 units up. ■ Use the description to write a quadratic function in vertex form. The function f (x) = x 2 is translated 1 unit right to create g. translation 1 unit right: h = 1 g (x) = a (x - h) 2 + k → g (x) = (x - 1) 2 5-1 Using Transformations to Graph Quadratic Functions (pp. 315–322) EXERCISES EXAMPLES Vocabulary absolute value of a complex number . . . . . . . . . . . . . . . . . 382 axis of symmetry . . . . . . . . . . . 323 binomial . . . . . . . . . . . . . . . . . . . 336 completing the square . . . . . . 342 complex conjugate . . . . . . . . . . 352 complex number . . . . . . . . . . . 351 complex plane . . . . . . . . . . . . . 382 discriminant . . . . . . . . . . . . . . . 357 imaginary number . . . . . . . . . . 350 imaginary part . . . . . . . . . . . . . 351 imaginary unit . . . . . . . . . . . . . 350 maximum value . . . . . . . . . . . . 326 minimum value . . . . . . . . . . . . 326 parabola . . . . . . . . . . . . . . . . . . . 315 quadratic function . . . . . . . . . . 315 quadratic inequality in two variables . . . . . . . . . . . . . . . . 366 quadratic model . . . . . . . . . . . . 376 quadratic regression . . . . . . . . 376 real part . . . . . . . . . . . . . . . . . . . 351 root of an equation . . . . . . . . . 334 standard form . . . . . . . . . . . . . . 324 trinomial. . . . . . . . . . . . . . . . . . . 336 vertex form. . . . . . . . . . . . . . . . . 318 vertex of a parabola . . . . . . . . . 318 zero of a function . . . . . . . . . . . 333
Transcript
392 Chapter 5 Quadratic Functions
Complete the sentences below with vocabulary words from the list above.
1. The number 5i can be classified as both a(n) −−−−−− ? and a −−−−−− ? .
2. The value of the input x that makes the output f (x) equal zero is called the −−−−−− ? .
3. The −−−−−− ? is the point at which the parabola intersects the axis of symmetry.
4. The type and number of solutions to a quadratic equation can be determined by finding the −−−−−− ? .
5. When a parabola opens upward, the y-value of the vertex is the −−−−−− ? of a quadratic function.
Graph each function by using a table.
6. f (x) = - x 2 - 2x 7. f (x) = 1 _ 2
x 2 + 3x - 4
Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function.
8. g (x) = 4 (x - 2) 2 9. g (x) = -2 (x + 1) 2
10. g (x) = 1 _ 3
x 2 - 3 11. g (x) = - (x + 2) 2 + 6
Use the description to write each quadratic function in vertex form.
12. f (x) = x 2 is reflected across the x-axis and translated 3 units down to create g.
13. f (x) = x 2 is vertically stretched by a factor of 2 and translated 4 units right to create g.
14. f (x) = x 2 is vertically compressed by a factor of 1 __ 4 and translated 1 unit left to create g.
■ Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph g (x) = 1 __
2 x 2 + 3.
g (x) = 1 __ 2 x 2 + 3 is f
vertically compressed by a factor of 1 __
2 and
translated 3 units up.
■ Use the description to write a quadratic function in vertex form. The function f (x) = x 2 is translated 1 unit right to create g.
translation 1 unit right: h = 1
g (x) = a (x - h) 2 + k → g (x) = (x - 1) 2
5-1 Using Transformations to Graph Quadratic Functions (pp. 315–322)
For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.
15. f (x) = x 2 - 4x + 3 16. g (x) = x 2 + 2x + 3
17. h (x) = x 2 - 3x 18. j (x) = 1 _ 2
x 2 - 2x + 4
Find the minimum or maximum value of each function.
19. f (x) = x 2 + 2x + 6 20. g (x) = 6x - 2 x 2
21. f (x) = x 2 - 5x + 1 22. g (x) = -2 x 2 - 8x + 10
23. f (x) = -x 2 - 4x + 8 24. g (x) = 3 x 2 + 7
■ For f (x) = - x 2 + 2x + 3, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.
a. Because a < 0, the e.
�
��
�
������
������ ������
������parabola opens downward.
b. axis of symmetry:
x = - b _ 2a
= - 2 _ 2(-1)
= 1
c. f (1) = - 1 2 + 2(1) + 3 = 4
The vertex is (1, 4).
d. Because c = 3, the y-intercept is 3.
5-2 Properties of Quadratic Functions in Standard Form (pp. 323–330)
EXERCISESE X A M P L E
Find the roots of each equation by factoring.
25. x 2 - 7x - 8 = 0 26. x 2 - 5x + 6 = 0
27. x 2 = 144 28. x 2 - 21x = 0
29. 4 x 2 - 16x + 16 = 0 30. 2 x 2 + 8x + 6 = 0
31. x 2 + 14x = 32 32. 9 x 2 + 6x + 1 = 0
Write a quadratic function in standard form for each given set of zeros.
33. 2 and -3 34. 1 and -1
35. 4 and 5 36. -2 and -3
37. -5 and -5 38. 9 and 0
■ Find the roots of x 2 + x = 30 by factoring.
x 2 + x - 30 = 0 Rewrite in standard form.
Factor.
Zero Product Property.Solve each equation.
(x - 5) (x + 6) = 0
x - 5 = 0 or x + 6 = 0 x = 5 or x = -6
■ Write a quadratic function with zeros 8 and -8. x = 8 or x = -8 Write zeros as solutions.
Set equations equal to 0.
Converse Zero Product Property
Replace 0 with f (x) .
x - 8 = 0 or x + 8 = 0
(x - 8) (x + 8) = 0
f (x) = x 2 - 64
5-3 Solving Quadratic Equations by Graphing and Factoring (pp. 333–340)
EXERCISESE X A M P L E S
Solve each equation by completing the square.
39. x 2 - 16x + 48 = 0 40. x 2 + 20x + 84 = 0
41. x 2 - 6x = 16 42. x 2 - 14x = 13
Write each function in vertex form, and identify its vertex.
Write a quadratic function that fits each set of points.
72. (-1, 8) , (0, 6) , and (1, 2)
73. (0, 0) , (1, -1) , and (2, -6)
Construction For Exercises 74–77, use the table of copper wire gauges.
Common U.S. Copper Wire Gauges
Gauge Diameter (in.)Resistance per 1000 ft (ohms)
24 0.0201 25.67
22 0.0254 16.14
20 0.0320 10.15
18 0.0403 6.385
74. Find a quadratic regression equation to model the diameter given the wire gauge.
75. Use your model to predict the diameter for a 12-gauge copper wire.
76. Find a quadratic regression equation to model the resistance given the wire gauge.
77. Use your model to predict the resistance for a 26-gauge copper wire.
■ Find a quadratic model for the wattage of fluorescent bulbs F given the comparable incandescent bulb wattage I. Use the model to estimate the wattage of a fluorescent bulb that produces the same amount of light as a 120-watt incandescent bulb.
Wattage Comparison
Incandescent (watts) 40 60 75 90 100
Fluorescent (watts) 11 15 20 23 28
Enter the data into two lists in a graphing calculator. Use the quadratic regression feature.
The model is F (I ) ≈ 0.0016 I 2 + 0.0481I + 6.48. A 36-watt fluorescent bulb produces about the same amount of light as a 120-watt incandescent bulb.
5-8 Curve Fitting with Quadratic Models (pp. 374–381)
EXERCISESE X A M P L E
Perform each indicated operation, and write the result in the form a + bi.
78. ⎜-3i⎟ 79. ⎜4 - 2i⎟
80. ⎜12 - 16i⎟ 81. ⎜7i⎟
82. (1 + 5i) + (6 - i) 83. (9 + 4i) - (3 + 2i)
84. (5 - i) - (11 - i) 85. -5i (3 - 4i)
86. (5 - 2i) (6 + 8i) 87. (3 + 2i) (3 - 2i)
88. (4 + i) (1 - 5i) 89. (-7 + 4i) (3 + 9i)
90. i 32 91. -5 i 21
92. 2 + 9i _ -2i
93. 5 + 2i _ 3 - 4i
94. 8 - 4i _ 1 + i
95. -12 + 26i _ 2 + 4i
Perform each indicated operation, and write the result in the form a + bi.
Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function.
1. g (x) = (x + 1) 2 - 2 2. h (x) = - 1 _ 2
x 2 + 2
3. Use the following description to write a quadratic function in vertex form: f (x) = x 2 is vertically compressed by a factor of 1 __
2 and translated 6 units right to create g.
For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.
4. f (x) = -x 2 + 4x + 1 5. g (x) = x 2 - 2x + 3
6. The area A of a rectangle with a perimeter of 32 cm is modeled by the function A (x) = - x 2 + 16x, where x is the width of the rectangle in centimeters. What is the maximum area of the rectangle?
Find the roots of each equation by using factoring.
7. x 2 - 2x + 1 = 0 8. x 2 + 10x = -21
Solve each equation.
9. x 2 + 4x = 12 10. x 2 - 12x = 25
11. x 2 + 25 = 0 12. x 2 + 12x = -40
Write each function in vertex form, and identify its vertex.
13. f (x) = x 2 - 4x + 9 14. g (x) = x 2 - 18x + 92
Find the zeros of each function by using the Quadratic Formula.
15. f (x) = (x - 1) 2 + 7 16. g (x) = 2 x 2 - x + 5
17. The height h in feet of a person on a waterslide is modeled by the function h (t) = -0.025 t 2 - 0.5t + 50, where t is the time in seconds. At the bottom of the slide, the person lands in a swimming pool. To the nearest tenth of a second, how long does the ride last?
18. Graph the inequality y < x 2 - 3x - 4.
Solve each inequality.
19. - x 2 + 3x + 5 ≥ 7 20. x 2 - 4x + 1 > 1
For Exercises 21 and 22, use the table showing the average cost of LCD televisions at one store.
21. Find a quadratic model for the cost of a television given its size.
22. Use the model to estimate the cost of a 42 in. LCD television.
Perform the indicated operation, and write the result in the form a + bi.
FOCUS ON SAT MATHEMATICS SUBJECT TESTSThe SAT Mathematics Subject Tests assess knowledge from course work rather than ability to learn. The Level 1 test is meant to be taken by students who have completed two years of algebra and one year of geometry, and it tests more elementary topics than the Level 2 test.
You may want to time yourself as you take this practice test. It should take you about 8 minutes to complete.
1. For what value of c will 3 x 2 - 2x + c = 0 have exactly one distinct real root?
(A) - 2 _ 3
(B) - 1 _ 3
(C) 0
(D) 1 _ 3
(E) 2 _ 3
2. If m and n are real numbers, i 2 = -1, and (m - n) - 4i = 7 + ni, what is the value of m?
(A) -4
(B) -3
(C) 1
(D) 3
(E) 4
3. If x 2 - 5x + 6 = (x - h) 2 + k, what is the value of k?
(A) - 25 _ 4
(B) - 5 _ 2
(C) - 1 _ 4
(D) 0
(E) 6
4. What is the solution set of y 2 - 2y ≤ 3y + 14?
(A) y ≥ -2
(B) y ≤ 7
(C) y ≤ -2 or y ≥ 7
(D) -7 ≤ y ≤ 2
(E) -2 ≤ y ≤ 7
5. Which of the following is a factor of (a - 1) 2 - b 2 ?
(A) a + b - 1
(B) a - b
(C) a - 1
(D) a - b + 1
(E) 1 - b
6. If z = 5 - 4i and i 2 = -1, what is z ?
(A) 1
(B) 3
(C) 9
(D) √ � 41
(E) √ � 42
You will need to use a calculator for some of the problems on the SAT Mathematics Subject Tests. Before test day, make sure that you are familiar with the features of the calculator that you will be using.
Multiple Choice: Work BackwardWhen taking a multiple-choice test, you can sometimes work backward to determine which answer is correct. Because this method can be time consuming, it is best used only when you cannot solve a problem in any other way.
Which expression is equivalent to 2x2- 3x - 14?
(2x + 7)(x + 2) (2x - 7)(x + 2)
(2x - 7)(x - 2) (2x + 7)(x - 2)
If you have trouble factoring the quadratic expression given in the question, you can multiply the binomials in the answer choices to find the product that is the same as 2x2
- 3x - 14.
Try Choice A: (2x + 7)(x + 2) = 2 x 2 + 11x + 14
Try Choice B: (2x - 7)(x - 2) = 2 x 2 - 11x + 14
Try Choice C: (2x - 7)(x + 2) = 2x2- 3x - 14
Choice C is the answer.
Note: Trying choice D can help you check your work.
What is the solution set of x 2 - 36 < 0?
x < -6 or x > 6 -36 < x < 36
-6 < x < 6 x < -36 or x > 36
If you have trouble determining the solution set, substitute values of x into the inequality. Based on whether the values make the inequality true or false, you may be able to eliminate one or more of the answer choices.
When x = 0, the inequality is true. Therefore, the solution set must include x = 0. Because choices F and J do not include x = 0, they can be eliminated.
Substitute 10 for x: x 2 - 36 < 0 → (10) 2 - 36 0 → 64 0 ✘
When x = 10, the inequality is false. Therefore, the solution set does not include x = 10. Because choice H includes x = 10, it can be eliminated.
The only remaining choice is choice G. Therefore, choice G must be correct.
Read each test item, and answer the questions that follow.
Item AWhat are the zeros of the function g (x) = 6 x 2 - 8x - 4, rounded to the nearest hundredth?
-10.32 and 2.32 1.72 and -0.39
-1.72 and 0.39 10.32 and -2.32
1. Rachel cannot remember how to determine the zeros of a quadratic function, so she plans to pick one of the answer choices at random. What could Rachel do to make a more educated guess?
2. Describe how to find the correct answer by working backward.
Item BA portable television has a screen with a diagonal of 4 inches. The length of the screen is 1 inch greater than its width. What are the dimensions of the screen to the nearest hundredth?
1.28 inches by 2.28 inches
1.28 inches by 3.28 inches
2.28 inches by 2.28 inches
2.28 inches by 3.28 inches
3. Can any of the answer choices be eliminated immediately? If so, which choices and why?
4. Describe how you can determine the correct answer by using the Pythagorean Theorem and working backward.
Item CWhich of the following is a solution of (x + 4) 2 = 25?
x = -9 x = 0
x = -1 x = 9
5. Explain how to use substitution to determine the correct answer.
6. Check whether choice A is correct by working backward. Explain your findings. What should you do next?
Item DThe height h of a golf ball in feet t seconds after it is hit into the air is modeled by h(t) = -16 t 2 + 64t. How long is the ball in the air?
2 seconds 12 seconds
4 seconds 16 seconds
7. The measurements given in the answer choices represent possible values of which variable in the function?
8. Describe how you can work backward to determine that choice F is not correct.
Item EThe base of a triangle is 4 in. longer than twice its height. If the triangle has an area of 24 in 2 , what is its height?
2 in. 6 in.
4 in. 8 in.
9. What equation do you need to solve to find the value of h?
10. Try choice A by working backward. Explain your findings. What should you do next?
You can also work backward to check whether the answer you found by another method is correct or reasonable.
2. Which of these functions does NOT have zeros at -1 and 4?
f (x) = x 2 - 3x - 4
f (x) = 2 x 2 + 6x - 8
f (x) = -x 2 + 3x + 4
f (x) = 2 x 2 - 6x - 8
3. Dawn and Julia are running on a jogging trail. Dawn starts running 5 minutes after Julia does. If Julia runs at an average speed of 8 ft/s and Dawn runs at an average speed of 9 ft/s, how many minutes after Dawn starts running will she catch up with Julia?
5 minutes 40 minutes
27 minutes 45 minutes
4. Which equation has intercepts at (20, 0, 0), (0, 40, 0), and (0, 0, 5)?
20x + 40y + 5z = 0
20x + 40y + 5z = 1
4x + 8y + z = 5
2x + y + 8z = 40
5. Which graph represents the function
f(x) = - 1 _ 2 (x - 3) - 4?
84
-4
8
4
y
-4-8 0
(-1, 6)(7, 2)
-8
4y
0
(-3, -4) (1, -6)
84-4
x
-8
-4
4y
0-8 84-4
(5, -5)
(-5, 0) x
4
-4
8y
-4-8 0
(5, 0)(-1, 3) x
6. What is the equation of the function graphed below?
-2
2
-4-6 0
(-1, 3)
2
x
y
(-3, -1)
y = (x - 3) 2 - 1 y = (x - 1) 2 - 3
y = (x + 3) 2 - 1 y = (x + 1) 2 - 3
KEYWORD: MB7 TestPrep
A211NLS_c05stp_0400_0401.indd 400A211NLS_c05stp_0400_0401.indd 400 2/3/10 11:33:38 AM2/3/10 11:33:38 AM
Cumulative Assessment, Chapters 1–5 401
7. If the relationship between x and y is quadratic, which value of y completes the table?
x -3 -1 1 3 5
y 21 7 27 61
3 9
7 17
8. Which is equivalent to the expression 5(6 - 8i )
_ 2 - i
?
-20 + 10i 15 - 40i
15 - 8i 20 - 10i
9. What is the inverse of the following matrix?
⎡ ⎢
⎣ -2 -4
4 2 ⎤ �
⎦
⎡
⎢
⎣
- 1 _
6
1 _ 3
- 1 _ 3
1 _ 6
⎤
�
⎦
⎡
⎢
⎣
1 _ 6
- 1 _ 3
1 _ 3
- 1 _ 6
⎤
�
⎦
⎡
⎢
⎣
- 1 _
2
1 _ 4
- 1 _ 4
1 _ 2
⎤
�
⎦
⎡
⎢
⎣ 2 -4
4 -2
⎤ �
⎦
Gridded Response 10. What value of x makes the equation
x 2 + 64 = 16x true?
11. The table shows the fees that are charged at an airport parking lot for various lengths of time. What is the slope of the linear function that models the parking fee f in dollars for h number of hours?
Time (h) 1 3 5 7
Parking Fee ($) 3.35 5.05 6.75 8.45
12. What is the x-value of the vertex of f(x) = 2 x 2 - 15x + 5?
13. What is the value of c given that the following system is dependent?
⎧ ⎨
⎩ 2y - x + 10 = 0
3x - 6y - c = 16
Short Response
14.
⎧ �
⎨ �
⎩
-4x + 8y - 2z = 8
4x - 4y + 2z = -5
x + 4y - 2z = 15
a. Write the augmented matrix that could be used to solve the system of equations given above.
b. Find the solution of the system, and explain how you determined your answer.
15. The graph below shows a feasible region for a set of constraints.
8642
2
4
6
8
0
y
x
a. Write the constraints for the feasible region.
b. Maximize the objective function P = 3x - 4y under these constraints.
16. Consider the function f (x) = x 2 - 2x - 48.
a. Determine the roots of the function. Show your work.
b. The function f is translated to produce the function g. The vertex of g is the point (3, 30) .Write the function rule for g in vertex form, and explain how you determined your answer.
Extended Response 17. A small alteration store charges $15.00 per hour
plus a $12.50 consulting fee for alterations. A competing store charges $20.00 per hour but does not charge a consulting fee.
a. For each store, write a linear function c that can be used to find the total cost of an alteration that takes h hours.
b. For which values of h is the small alteration store less expensive than the competing store? Explain how you determined your answer.
c. The small store wants to adjust its pricing so that it is less expensive than the competing store for any alteration job that takes an hour or more. By how much should the small store lower its consulting fee in order to make this adjustment?
In nearly all standardized tests, you cannot enter a negative value as the answer to a gridded-response question. If you get a negative value as an answer to one of these questions, you have probably made a mistake in your calculations.
Complete the sentences below with vocabulary words from the list above.
1. A(n) −−−−−− ? has a common difference, and a(n) −−−−−− ? has a common ratio.
2. A series that has no limit −−−−−− ? , whereas a series that approaches a limit −−−−−− ? .
3. A(n) −−−−−− ? defines the nth term. A(n) −−−−−− ? defines the next term by using one or more of the previous terms.
4. A(n) −−−−−− ? continues without end, and a(n) −−−−−− ? has a last term.
5. Each step in a repeated process is called a(n) −−−−−− ? .
Find the first 5 terms of each sequence.
6. a n = n - 9 7. a n = 1 _ 2
n 2
8. a n = (- 3 _ 2
) n -1
9. a 1 = 55 and a n = a n -1 - 2
10. a 1 = 200 and a n = 1 _ 5
a n -1
11. a 1 = -3 and a n = -3 a n -1 + 1
Write a possible explicit rule for the nth term of each sequence.
12. -4, -8, -12, -16, -20, . . .
13. 5, 20, 80, 320, 1280, . . .
14. -24, -19, -14, -9, -4, . . .
15. 27, 18, 12, 8, 16 _ 3
, . . .
16. Sports Suppose that a basketball is dropped from a height of 3 ft. If the ball rebounds to 70% of its height after each bounce, how high will the ball reach after the 4th bounce? the 9th bounce?
■ Find the first 5 terms of the sequence with a 1 = -52; a n = 0.5 a n -1 + 2.
Evaluate the rule using each term to find the next term.
n 1 2 3 4 5
a n -52 -24 -10 -3 0.5
■ Write an explicit rule for the nth term of 100, 72, 44, 16, -12, . . . .
Examine the differences or ratios. Terms 100 72 44 16 -12
1st differences 28 28 28 28
The first differences are constant, so the sequence is linear.
The first term is 100, and each term is 28 less than the previous term.
24. Finance A household has a monthly mortgage payment of $1150. How much is paid by the household after 2 years? 15 years?
■ Expand ∑ k = 1
5
(-1) n+1 (11 - 2n) , and evaluate.
∑ k = 1
5
(-1) n+1 (11 - 2n) = (-1) 2 (11 - 2)
+ (-1) 3 (11 - 4) + (-1) 4 (11 - 6)
+ (-1) 5 (11 - 8) + (-1) 6 (11 - 10)
= 9 - 7 + 5 - 3 + 1
= 5 Simplify.
■ Evaluate ∑ k = 1
8
k 2 .
Use summation formula for a quadratic series.
∑k = 1
8
k2 =n (n + 1) (2n + 1) __
6
=8 (8 + 1) (2 · 8 + 1) __
6=
72 (17) _6
= 204
12-2 Series and Summation Notation (pp. 870–877)
EXERCISESE X A M P L E S
Find the 11th term of each arithmetic sequence.
25. 23, 19, 15, 11, . . . 26. 1 _ 5
, 3 _ 5
, 1, 7 _ 5
, 9 _ 5
, . . .
27. -9.2, -8.4, -7.6, -6.8, . . .
28. a 3 = 1.5 and a 4 = 5
29. a 6 = 47 and a 8 = 21
30. a 5 = -7 and a 9 = 13
Find the indicated sum for each arithmetic series.
31. S 18 for -1 - 5 - 9 - 13 + �
32. S 12 for 1 _ 3
+ 1 _ 6
+ 0 - 1 _ 6
+ �
33. ∑ k = 1
15
(-14 + 3k)
34. ∑ k = 1
15
( 3 _ 2
k + 10)
35. Savings Kelly has $50 and receives $8 a week for allowance. He wants to save all of his money to buy a new mountain bicycle that costs $499. Write an arithmetic sequence to represent the situation. Then find whether Kelly will be able to buy the new bicycle after one year (52 weeks) .
■ Find the 12th term for the arithmetic sequence 85, 70, 55, 40, 25, . . ..
Find the common difference: d = 70 - 85 = -15.
a n = a 1 + (n - 1) d General rule
a 12 = 85 + (12 - 1) (-15) Substitute.
= -80 Simplify.
■ Find ∑ k = 1
11
(-2 - 33k) .
Find the 1st and 11th terms.
a 1 = -2 -33 (1) = -35
a 11 = -2 - 33 (11) = -365
Find S 11 .
S n = n ( a 1 + a n
_ 2
) Sum formula
S 11 = 11 ( -35 - 365 _ 2
) Substitute.
= -2200
12-3 Arithmetic Sequences and Series (pp. 879–887)
Find the 9th term of the geometric sequence with the given terms.
40. a 3 = 24 and a 4 = 96
41. a 1 = 2 _ 3
and a 2 = - 4 _ 3
42. a 4 = -1 and a 6 = -4
43. a 3 = 4 and a 6 = 500
Find the geometric mean of each pair of numbers.
44. 10 and 2.5 45. 1 _ 2
and 8
46. √ � 3
_ 96
and √ � 3
_ 6
47. 5 _ 12
and 125 _ 108
Find the indicated sum for each geometric series.
48. S 5 for 1 + 1 _ 3
+ 1 _ 9
+ 1 _ 27
+ �
49. S 6 for - 4 _ 5
+ 8 - 80 + 800 + �
50. ∑ k = 1
8
(4) k -1
51. ∑ k = 1
7
-2 (5) k -1
52. ∑ k = 1
6
60 (- 1 _ 2
) k -1
53. ∑ k = 1
5
18 ( 1 _ 2
) k -1
54. Depreciation A new photocopier costs $9000 and depreciates each year such that it retains only 65% of its preceding year’s value. What is the value of the photocopier after 5 years?
55. Rent A one-bedroom apartment rents for $650 a month. The rent is expected to increase by 6% per year.
a. What will be the annual rent expense on the apartment after 5 years?
b. What will be the total amount spent on rent if a person rents the apartment for the entire 5-year period?
■ Find the 8th term of the geometric sequence 6, 24, 96, 384, . . ..
Find the common ratio. r = 24 _ 6
= 4
Write a rule, and evaluate for n = 8.
a n = a 1 r n -1 General rule
a 8 = 6 (4) 8 -1 = 98,304
■ Find the 8th term of the geometric sequence with a 4 = -1000 and a 6 = -40.
Step 1 Find the common ratio.
a 6 = a 4 r (6 -4) Use the given terms.
-40 = -1000 r 2 Substitute.
1 _ 25
= r 2 Simplify.
± 1 _ 5
= r
Step 2 Find a 1 using both possible values for r.
-1000 = a 1 (
1 _ 5
)
4-1
or -1000 = a 1 (
- 1 _ 5
)
4-1
a 1 = -125,000 or a 1 = 125,000
Step 3 Write the rule and evaluate for a 8 by using both possible values for r.
Find the sum of each infinite series, if it exists.
56. -2700 + 900 - 300 + 100 + ⋯
57. -1.2 - 0.12 - 0.012 - 0.0012 + ⋯
58. -49 - 42 - 36 - 216_7
+ ⋯
59. 4 + 4_5
+ 4_25
+ 4_125
+ ⋯
60. k = 1
∞
9_ 3 k
61. k = 1
∞
-7 (3_5)k
62. k = 1
∞
(-1)k +1( 1_ 8 k )
63. k = 1
∞
(4_3)k
Use mathematical induction to prove each statement.
64. 2 + 4 + 8 + ⋯ + 2 n = 2 n +1 - 2
65. 1 + 5 + 25 + ⋯ + 5 n-1 = 5 n - 1 _ 4
66. 1 _ 3
+ 1 _ 15
+ ⋯ + 1 _ 4 n 2 - 1
= n_2n + 1
67. Recreation A child on a swing is let go from a vertical height so that the distance that he travels in the first back-and-forth swing is exactly 9 feet.
a. If each swing decreases the distance by 85%, write an infinite geometric series that expresses the distance that the child travels in feet.
b. What is the total distance that the child in the swing travels before the swing stops?
Find the sum of each infinite series, if it exists.
■ -9261 + 441 - 21 + 1 + ⋯
r = 441_-9261
= - 1_21
Converges: ⎜r⎟ < 1
S = a 1 _
1 - rSum formula
= -9261_
1 - (- 1 ___ 21
) = -9261_
22 __ 21
= -194,481_
22 , or -8840.0
45
■ k = 1
∞
-5 ( 7_10)k -1
= -5 - 35_10
- 245_100
+ ⋯ Evaluate.
r = - 35 __
10 _
-5= 7_
10Converges: ⎜r⎟ < 1
S = a 1 _
1 - r = -5 _
1 - 7 ___ 10
= -5 _
3 __ 10
= - 50 _
3 , or -16.
6
■ Use mathematical induction to prove
2 + 5 + ⋯ + (3n - 1) = n _ 2
(3n + 1) .
Step 1 Base case: Show that the statement is true for n = 1.
2 = n _ 2
(3n + 1) = 1 _ 2
(3 · 1 + 1) = 2 True
Step 2 Assume that the statement is true for a natural number k.
2 + 5 + ⋯ + (3k - 1) = k _ 2
(3k + 1) Replace n with k.
Step 3 Prove that it is true for the natural number k + 1.
Write a possible explicit rule for the nth term of each sequence.
3. -4, -2, 0, 2, 4, . . . 4. 54, 18, 6, 2, 2 _ 3
, . . .
Expand each series and evaluate.
5. k = 1
4
5 k 3 6. k = 1
7
(-1) k+1 (k)
Find the 9th term of each arithmetic sequence.
7. -19, -13, -7, -1, . . . 8. a 2 = 11.6 and a 5 = 5
9. Find 2 missing terms in the arithmetic sequence 125, , , 65.
Find the indicated sum for each arithmetic series.
10. S 20 for 4 + 7 + 10 + 13 + . . . 11. k = 1
12
(-9k + 8)
12. The front row of a theater has 16 seats and each subsequent row has 2 more seats than the row that precedes it. How many seats are in the 12th row? How many seats in total are in the first 12 rows?
Find the 10th term of each geometric sequence.
13. 3 _ 256
, 3 _ 64
, 3 _ 16
, 3 _ 4
, . . . 14. a 4 = 2 and a 5 = 8
15. Find the geometric mean of 4 and 25.
Find the indicated sum for each geometric series.
16. S 6 for 2 + 1 + 1 _ 2
+ 1 _ 4
+ . . . 17. k = 1
6
250 (- 1 _ 5
) k-1
18. You invest $1000 each year in an account that pays 5% annual interest. How much is the first $1000 you invested worth after 10 full years of interest payments? How much in total do you have in your account after 10 full years?
Find the sum of each infinite geometric series, if it exists.
19. 200 - 100 + 50 - 25 + . . . 20. k = 1
∞
2 ( 7 _ 8
) k
Use mathematical induction to prove 1 _ 2
+ 3 _ 2
+ 5 _ 2
+ ⋯ + 2n - 1 _ 2
= n 2 _ 2
.
21. Step 1
22. Step 2
23. Step 3
a207se_c12cht_0916.indd 916 11/13/05 6:29:37 PM
College Entrance Exam Practice 917
FOCUS ON SATWhen you get your SAT scores, you are given the percentile in which your scores fall. This tells you the percentage of students that scored lower than you did on the same test. You’ll see your percentile score at the national and state levels. They are usually not the same.
You may want to time yourself as you take this practice test. It should take you about 8 minutes to complete.
1. The first term of a sequence is 6, and each successive term is 3 less than twice the preceding term. What is the sum of the first four terms of the sequence?
(A) 27
(B) 30
(C) 51
(D) 57
(E) 123
2. The first term of a sequence is 2, and the nth term is defined to be 3n - 1. What is the average of the 7th, 10th, and 12th terms?
(A) 24.5
(B) 28
(C) 29
(D) 32
(E) 84
3. The first term of an arithmetic sequence is -5. If the common difference is 4, what is the 7th term of the sequence?
(A) -20,480
(B) -29
(C) 19
(D) 20
(E) 23
4. A population of 50 grows exponentially by doubling every 4 years. After how many years will the population have 1600 members?
(A) 20
(B) 16
(C) 10
(D) 6
(E) 5
5. Which of the following sequences can be expressed by the rule a n = n - 1 ____
n + 1 ?
(A) 3, 2, 5 _ 3
, 3 _ 2
, 7 _ 5
, . . .
(B) 1 _ 2
, 3 _ 4
, 5 _ 6
, 7 _ 8
, 9 _ 10
, . . .
(C) 1 _ 3
, 2 _ 4
, 3 _ 5
, 4 _ 6
, 5 _ 7
, . . .
(D) 0, 1 _ 3
, 1 _ 2
, 3 _ 5
, 2 _ 3
, . . .
(E) 0, 3 _ 2
, 3, 5 _ 3
, 5, . . .
6. Which of the following sequences is a geometric sequence?
(A) -7, 14, -28, 56, -112, . . .
(B) -4, -6, -8, -10, -12, . . .
(C) -3, 1, -3, 1, -3, . . .
(D) 4, 12, 48, 144, 576, . . .
(E) 1, 4, 9, 16, 25, . . .
Read each problem carefully, and make sure that you understand what the question is asking. Before marking your final answer on the answer sheet, check that your answer makes sense in the context of the question.
a207se_c12cep_0917.indd 917 11/13/05 6:28:48 PM
Simple interest formula
918 Chapter 12 Sequences and Series
Short Response Ariana is saving money for a new car. She saves $40 the first week, $45 the second week, $50 the third week, and so on. Explain whether an arithmetic or geometric sequence would best represent this situation. Use a sequence or series to determine the amount that she will save in the 8th week and the total amount that she will have saved after 8 weeks.
Create an outline for your response.
Follow the outline, and write out your response.
Short/Extended Response: Outline Your ResponseAnswering short and extended response items on tests is a lot like writing essays in English class. You can use an outline to plan your response to the question. Outlines help you organize the main points and the order in which they will appear in your answer. Outlining your response will help ensure that your explanation is clearly organized and includes all necessary information.
An arithmetic sequence would best represent this situation because Ariana is adding $5 each week to the amount that she saves. This would be an arithmetic sequence where the first term is 40 and the common difference is 5.
The sequence for the amount saved each week is a n = 40 + 5 (n - 1) . The series for the total amount
saved is ∑ k = 1
n
[40 + 5 (k - 1) ] .
The amount saved in the 8th week is a 8 = 40 + 5 (8 - 1) = $75.
The total saved after 8 weeks is ∑ k = 1
8
[40 + 5 (k - 1) ] = $460.
Clearly indicate which is the sequence and which is the series for the second step.
Include evidence to explain the answer for the first step.
Show how you found the answers for the last two steps.
Outline
1. Explain whether arithmetic or geometric.2. Write sequence and series.3. Find amount saved in 8th week.4. Find total saved after 8 weeks.
When you finish your response, check it against your outline to make sure that you did not leave out any details.
Read each test item and answer the questions that follow.
Item AShort Response Explain how to determine whether an infinite geometric series has a sum.
1. What should be included in an outline of the response for this test item?
2. Read the two different outlines below. Which outline is the most useful? Why?
Student A I. Definition of an infinite geometric series
and the common ratio r.
II. Definition of the sum of an infinite geometric series.
III. Explain for which values of r that a sum exists.
Student BA. Geometric series has a common ratio.
B. Common ratio has to be less than 1.
Item B Extended Response A pattern for stacking cereal boxes is shown at right.a. Explain how many
boxes are in a 9-row display.
b. If 91 boxes are to be stacked in this display, explain how many rows the display will have.
3. Read the outline below. Identify any areas that need improvement. Rewrite the outline to make it more useful.
Outline 1. Find the number of boxes if there are
9 rows.
2. Find the number of rows needed for 91 boxes.
Item C Extended Response A pattern of squares is created by doubling the dimensions of the previous square. Explain how to find the sum of the perimeters of the first 8 squares if the first square is 5 cm wide.
4. A student correctly gave the following response. Write an outline for a response to this question.
To find the perimeters of the first 8 squares, I need to determine the series.
4 (5) + 4 (10) + 4 (20) + 4 (40) + �
4 (5 + 10 + 20 + 40 + �)
4 · 5 (1 + 2 + 4 + 8 + �)
20 (1 + 2 + 4 + 8 + �)
20 ∑ n = 1
8
2 (n-1)
Now that I know that the first term is 20 and the common ratio is 2, I can use the formula
S n = t 1 ( 1 - r n _ 1 - r
) , where t 1 is the first term
and r is the common ratio.
S n = 20 ( 1 - 2 8 _ 1 - 2
)
= 20 ( 1 - 256 _ -1
)
= 20 ( -255 _ -1
)
= 20 (255) = 5100
So the sum of the perimeters of the first 8 squares is 5100 cm.
3. Let f (x) = x 3 + 2 x 2 - 5x - 9. Which function would show f (x) reflected across the y-axis?
g (x) = - x 3 - 2 x 2 + 5x + 9
g (x) = - x 3 + 2 x 2 + 5x - 9
g (x) = 2 x 3 + 4 x 2 - 10x - 18
g (x) = x 3 + 2 x 2 - 5x - 5
4. Which function shows exponential decay?
f (x) = -5x
f (x) = 2.3 (6.7)x
f (x) = 0.49 (7.9)x
f (x) = 5.13 (0.32)x
5. A ball is dropped from a height of 10 feet. On each bounce, the ball bounces 60% of the height of the previous bounce. Which expression represents the height in feet of the ball on the nth bounce?
10 (0.6n)
10 (0.6)n-1
10 - n_0.6
10 (0.6)n
6. Which is the graph of the inequality 6x + 3y ≥ 9 x 2 - 3?
x
y
0
x
y
0
x
y
0
x
y
0
7. Gina opened a new deli. Her revenues for the first 4 weeks were $2000, $2400, $2880, and $3456. If the trend continues, which is the best estimate of Gina’s revenues in the 6th week?
$3856
$4032
$4147
$4980
8. What is the 9th term in the sequence?
a n = 1 _ 2 ( 2 n-1 ) + 4
36 132
68 260
9. Find the inverse of f (x) = 4x - 5.
f -1 (x) = -4x + 5
f -1 (x) = 1 _ 4 x + 5
f -1 (x) = x + 5 _ 4
f -1 (x) = 5x - 4
920 Chapter 12 Sequences and Series
A211NLS_c12stp_0920-0921.indd 920A211NLS_c12stp_0920-0921.indd 920 8/13/09 1:24:44 AM8/13/09 1:24:44 AM
10. What transformation has been applied to f to get g?
42-4 -2
4
2
y
x
g
f
-4
Horizontal compression by 1 _ 5
Horizontal stretch by 5
Vertical compression by 1 _ 3
Vertical stretch by 5
11. Find the zeros of f (x) = 2 x 2 + 5x - 12.
-4, 3 _ 2
-2, 3
- 3 _ 2 , 4
3 _ 2
, 2
Gridded Response 12. Find the common ratio of the geometric
sequence.
125, 50, 20, 8, . . .
13. A card is drawn from a deck of 52. What is the probability of drawing a 10 or a diamond, to the nearest hundredth?
14. What is the sum of the arithmetic series?
∑ k = 1
8
(7k - 3)
15. What is the y-value of the point that represents the solution to the given system of equations, to the nearest hundredth?
⎧ ⎨
⎩ 2y - 2 = 4x
6 - x = 8y
Short Response 16. Use the function f (x) = 3 √ � 5x to answer the
following questions.
a. What is the domain and range?
b. What is the inverse of f (x) ?
c. What is the domain and range of the inverse function?
d. Graph f (x) and f -1 (x) on the same coordinate plane.
17. Use the infinite geometric series ∑n = 1
∞
5____ 4 n - 1
to answer the following questions.
a. Determine if the series converges or diverges.
b. Find the sum of the infinite series, if it exists.
18. A grocery store display contains 3 cans on the top row and an additional can in each row forming a triangular shape.
a. Would you use a sequence or a series to represent the number of cans in the nth row? Explain.
b. How many cans are in the 12th row?
c. What does the series ∑k = 1
n
(k + 2) represent? Explain.
Extended Response 19. A test to be on a trivia show has two parts. 60%
of contestants pass the first part, and 20% pass the second part.
a. Draw a tree diagram that gives the probabilities for a contestant’s possible outcomes on the test.
b. If a contestant must pass both parts of the test to be on the show, how many contestants out of a group of 50 would likely make the show? Show your work.
c. Is it more likely that a contestant would pass both parts or fail both parts of the test? Explain.
In Item 11, you may choose to graph, factor, complete the square, or use the Quadratic Formula to find the zeros.
Cumulative Assessment, Chapters 1–12 921
A211NLS_c12stp_0920-0921.indd 921A211NLS_c12stp_0920-0921.indd 921 8/13/09 1:25:17 AM8/13/09 1:25:17 AM
Classify each system and determine the number of solutions.
10. ⎧ ⎨
⎩ y = x - 7
x + 9y = 16
11. ⎧ �
⎨ �
⎩ 1 _ 2
x + 2y = 3
x + 4y = 6
12. ⎧ ⎨
⎩ 5x - 10y = 8
x - 2y = 4
13. ⎧ ⎨
⎩ 4x - 3y = 21
2x - 2y = 10
14. Security A locksmith charges $25 to make a house call and $15 for each lock that is re-keyed. Another locksmith charges $10 to make a house call and $20 for each lock that is re-keyed. For how many locks will the total costs be the same?
■ Solve ⎧
⎨
⎩
x + y = 3
3x - 6y = -9 by using a graph and
a table.
Solve each equation for y.
⎧ �
⎨
�
⎩ y = -x + 3
y = 1 _
2 x + 3 _
2
Make a table of values. Graph the lines.
y = -x + 3 y = 1 _ 2
x + 3 _ 2
x y
0 3
1 2
4 1
x y
0 1.5
1 2
4 2.5
The solution is (1, 2).
3-1 Using Graphs and Tables to Solve Linear Systems (pp. 182–189)
EXERCISESE X A M P L E S
Complete the sentences below with vocabulary words from the list above.
1. A consistent and −−−−−− ? system has infinitely many solutions.
2. −−−−−− ? involves adding or subtracting equations to get rid of one of the variables in a system.
3. In a linear programming problem, the solution to the −−−−−− ? can be graphed as a(n) −−−−−− ? .
4. Each point in a(n) −−−−−− ? can be represented by a(n) −−−−−− ? .
5. A(n) −−−−−− ? system is a set of equations or inequalities that has at least one solution.
Use substitution to solve each system of equations.
15. ⎧ ⎨
⎩ y = 3x
2x - 3y = -7
16. ⎧ ⎨
⎩ y = x - 1
4x - y = 19
17. ⎧ ⎨
⎩ 4x - y = 0
6x - 3y = 12
18. ⎧ ⎨
⎩ 5x = -10y
8x - 4y = 40
Use elimination to solve each system of equations.
19. ⎧ ⎨
⎩ 4x + 5y = 41
7x + 5y = 53
20. ⎧ ⎨
⎩ -4x - y = -16
-4x - 5y = -32
21. ⎧ ⎨
⎩ 2x - y = 8
x + 2y = 9
22. ⎧ ⎨
⎩ 9x - 5y = 13
4x - 6y = 2
23. Mixtures A popular mixture of potpourri includes pine needles and lavender. If pine needles cost $1.50 per ounce and lavender costs $4.00 per ounce, how much of each ingredient should be mixed to make 80 oz of the potpourri that is worth $200?
■ Use substitution to solve ⎧
⎨
⎩
y = x + 6
4x - 5y = -18
.
4x - 5 (x + 6) = -18 Substitute for y. 4x - 5x - 30 = -18 ➔ x = -12
Substitute the x-value into either equation.
y = x + 6 ➔ y = (-12) + 6 ➔ y = -6
The solution to the system is (-12, -6) .
■ Use elimination to solve ⎧
⎨
⎩
7x - 2y = 2
3x + 4y = 30
.
Multiply the first equation by 2 to eliminate y.
⎧ ⎨
⎩ 7x - 2y = 2
3x + 4y = 30
➔ 2(7x - 2y = 2)
2(3x + 4y = 30
➔ 14x - 4y = 4
13x + 4y = 30
Add the equations. 17x = 34First part of the solution x = 2
Substitute the x-value into either equation.
3x + 4y = 30 ➔ 3 (2) + 4y = 30
➔ y = 6 Second part of the solution
The solution to the system is (2, 6) .
3-2 Using Algebraic Methods to Solve Linear Systems (pp. 190 –197)
EXERCISESE X A M P L E S
Graph each system of inequalities.
24. ⎧ ⎨
⎩ y + 1 > 4x
y ≤ x + 1
25. ⎧ ⎨
⎩ y - 3x < 3
3y ≥ x + 3
Graph the system of inequalities and classify the figure created by the solution region.
26.
⎧ �
⎨
� ⎩
y ≤ -x + 2
x > -1
y > -1
27.
⎧ �
�
⎨ � �
⎩
y ≥ 2x
y < 4
y > 2
y ≤ 1 _
2 x + 4
28. Business A coffee shop wants to make a maximum of 120 lb of a coffee mixture that costs less than $10/lb. The shop will mix coffee that is sold at $8/lb with coffee sold at $11.50/lb. Write and graph a system of inequalities that shows the possible mixtures of the two coffee types.
■ The combined annual sales for a company’s two divisions was almost $12 million. One of the divisions accounted for at least 75% of the total sales. Write and graph a system of inequalities that can be used to determine the possible combinations of sales for both divisions of the company.
Let x be one division, and let y be the other division with 75% of the sales.
Write the system of inequalities.
⎧ ⎨
⎩ x + y < 12
y ≥ 0.75(x + y)
➔ ⎧ ⎨
⎩ x + y < 12
y ≥ 3x
Graph the boundary lines, and shade accordingly. Notice also that x > 0 and y > 0.
The overlapping region is the solution for the system.
3-3 Solving Systems of Linear Inequalities (pp. 199–204)
33. Maximize P = 6x + 10y for the constraints from Exercise 29.
34. Minimize P = 14x + 9y for the constraints from Exercise 30.
Manufacturing A shoe insole company produces two models of insoles: an extra thick insole for sports shoes and a thinner insole for dress shoes. The thick insole requires 6 min of manufacturing time and generates a profit of $8. The thin insole requires 4 min of manufacturing time and generates a profit of $9. The manufacturing line runs at most 12 h a day, or 720 min. Because of demand, the company manufactures at least twice as many thick insoles as thin insoles.
35. Write the constraints, and graph the feasible region.
36. Write the objective function for the company’s profit.
37. What is the maximum profit that can be generated in one day?
38. Sales Each day, a cell phone stand sells between 10 and 25 cell phones with new service contracts, and between 5 and 10 cell phones without contracts. The stand never sells more than 30 new cell phones per day. The cell phone stand makes a commission of $35 for each phone with a contract and $5 for each phone without a contract. How many of each option would maximize the stand’s profit?
■ A café sells cold sandwiches and hot entrées. The range of items sold is shown in the table. The café has never sold more than a total of 125 sandwiches and entrées in one day. If the café makes a profit of $0.75 on each sandwich and $1 on each hot entrée, how many of each item would maximize the café profit?
Menu ItemMinimum
SoldMaximum
Sold
Cold sandwiches
60 80
Hot entrées 40 60
Let x be the number of cold sandwiches, and let y be the number of hot entrées.
Write the constraints.
⎧ ⎪
⎨
⎪ ⎩
60 ≤ x ≤ 80
40 ≤ y ≤ 60
x + y < 125
Graph the feasible region and identify vertices.
The feasible region has five vertices at (60, 40) , (60, 60) ,
(65, 60) , (80, 45) ,
and (80, 40) .
Write the objective function.
The objective function is P = 0.75x + y. P ( 0, 0) = 18 (0) + 25 (0) = 0
Evaluate the objective function at each vertex.
P(60, 40) = 0.75(60) + 40 = 85
P(60, 60) = 0.75(60) + 60 = 105
P(65, 60) = 0.75(65) + 60 = 108.75
P(80, 45) = 0.75(80) + 45 = 105
P(80, 40) = 0.75(80) + 40 = 100
The objective function is maximized at (65, 60). The maximum profit of $108.75 is obtained when 65 cold sandwiches and 60 hot entrées are sold.
Graph each linear equation in three-dimensional space.
43. x - 3y + 2z = 6 44. 2x - 4y - 2z = 4
45. -x + y - 5z = 5 46. 3x + 2y + z = -6
47. Consumer Economics Lee has $35 to purchase a combination of drinks, pizza, and ice cream for a party. Each drink costs $2, each pizza costs $9, and each quart of ice cream costs $4. Write a linear equation in three variables to represent this situation.
■ Graph (2, -1, 3) in three-dimensional space.
From the origin, move 2 units forward along the x-axis, 1 unit left, and 3 units up.
■ Graph the linear equation 3x + 6y - z = -6 in three-dimensional space.
Find the intercepts.
x-intercept: 3x = -6 ➔ x = -2
y-intercept: 6y = -6 ➔ y = -1
z-intercept: -z = -6 ➔ z = 6
Plot the points (-2, 0, 0), (0, -1, 0), and (0, 0, 6). Sketch a plane through the three points.
3-5 Linear Equations in Three Dimensions (pp. 214–218)
EXERCISESE X A M P L E S
Use elimination to solve each system of equations.
48.
⎧ ⎪
⎨ ⎪
⎩
x + 3y + 2z = 13
2x + 2y - z = 3
x - 2y + 3z = 6
49.
⎧ ⎪
⎨
⎪ ⎩
x + y + z = 2
3x + 2y - z = -1
3x - y = 4
Classify each system as consistent or inconsistent, and determine the number of solutions.
50.
⎧ ⎪
⎨
⎪ ⎩
x + y + z = -2
-x + 2y - 5z = 4
3x + 3y + 3z = 5
51.
⎧ ⎪
⎨ ⎪
⎩
-x - y + 2z = -3
4x + 4y - 8z = 12
2x + y - 3z = -2
■ Use elimination to solve ⎧ ⎪
⎨
⎪
⎩
3x + 2y - z = -1
x + 3y - z = -10
2x - y - 3z = -3
.
First, eliminate z to obtain a 2-by-2 system.
3x + 2y - z = -1
3x + 3y - z = -10
3(x + 3y - z = -10)
32x - y - 3z = -3
2x - y = 9 x + 10y = -27
The resulting 2-by-2 system is ⎧ ⎪
⎨
⎪ ⎩ 2x - y = 9
x + 10y = -27
.
Eliminate x.
( 2x - y = 9
-2 (x + 10y = -27 ) -21y = 63 ➔ y = -3
Substitute to solve for x and then z.
2x - y = 9 ➔ 2x - (-3) = 9 ➔ x = 3
3x + 2y - z = -1 ➔ 3(3) + 2(-3) - z = -1 ➔ z = 4
The solution to the system is (3, -3, 4).
3-6 Solving Linear Systems in Three Variables (pp. 220–226)
Classify each system and determine the number of solutions.
4. �
� 6y = 9x
8x + 4y = 20
5. �
� 12x + 3y = -9
-y - 4x = 3
6. �
� 3x - 9y = 21
6 = x - 3y
Use substitution or elimination to solve each system of equations.
7. �
� y = x - 2
x + 5y = 20
8. �
� 5x - y = 33
7x + y = 51
9. �
� x + y = 5
2x + 5y = 16
Graph each system of inequalities.
10. �
� 2y - 4x ≥ 4
y - x ≥ 1
11. �
� x + y ≥ 3
y - 4 ≤ 0
12. Chemistry A chemist wants to mix a new solution with at least 18% pure salt. The chemist has two solutions with 9% pure salt and 24% pure salt and wants to make at most 250 mL of the new solution. Write and graph a system of inequalities that can be used to find the amounts of each salt solution needed.
13. Minimize the objective function P = 5x + 9y under the following constraints. �
�
x ≥ 0
y ≥ 0
y ≤ 2x + 1
y ≤ -3x + 6
Graph each point in three-dimensional space.
14. (2, -1, 3) 15. (0, -1, 3) 16. (-2, 1, -1)
Business Use the following information and the table for Problems 17 and 18.A plumber charges $50 for repairing a leaking faucet, $150 for installing a sink, and $200 for an emergency situation. The plumber’s total income was exactly $1000 for each day shown in the table.
17. Write a linear equation in three variables to represent this situation.
18. Complete the table for the possible numbers of tasks each day.
Solve each system of equations using elimination, or state that the system is inconsistent or dependent.
FOCUS ON SAT MATHEMATICS SUBJECT TESTSIn addition to the SAT, the SAT Mathematics Subject Tests are required by some colleges for admission. Colleges that don’t require the SAT Mathematics Subject Tests may still use the scores to learn about your academic background and possibly place you in the appropriate college math class.
You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete.
Take the SAT Mathematics Subject Tests while the subject matter is fresh in your mind. You are not expected to be familiar with all the content covered on the tests, but you should have completed at least three years of college-prep math.
1. Which of the following systems of equations is represented by the graph?
(A)
⎧ � ⎨ �
⎩ y = -2x + 4
y = 1 _
4 x + 2
(B)
⎧ � ⎨ �
⎩ y = 2x - 4
y =- 1 _
4 x - 2
(C)
⎧ � ⎨ �
⎩ y = 2x + 4
y = 1 _
4 x - 2
(D)
⎧ � ⎨ �
⎩ y = 1 _
2 x + 4
y = 4x - 2
(E)
⎧ � ⎨ �
⎩ y = 1 _
2 x - 4
y = 4x + 2
2. If x - 2y = 1 and 2x - y = -4, then x + y = ?
(A) -9
(B) -7
(C) -5
(D) -3
(E) -1
3. In a fruit salad, there are two more bananas than apples and eight times as many cherries as apples. If a total of 22 pieces of fruit are used, how many of each type are in the salad?
(A) 2 apples, 4 bananas, 18 cherries
(B) 2 apples, 4 bananas, 16 cherries
(C) 2 apples, 0 bananas, 20 cherries
(D) 4 apples, 2 bananas, 12 cherries
(E) 4 apples, 8 bananas, 32 cherries
4. Which of the following inequalities is NOT graphed in the figure?
(A) y > -3x + 2
(B) 2y ≤ x - 6
(C) 0.5x ≥ y + 3
(D) 3x + y ≥ 2
(E) 6x + 2y > 4
5. If r = 3s + 1 and t = s - 4, then what is r in terms of t ?
