Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (2024)

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Big Ideas Math Book Algebra 2 Answer Key Chapter 1 Linear Functions

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  • Linear Functions Maintaining Mathematical Proficiency – Page 1
  • Linear Functions Maintaining Mathematical Practices – Page 2
  • Lesson 1.1 Parent Functions and Transformations – Page (4-10)
  • Parent Functions and Transformations 1.1 Exercises – Page (8-10)
  • Lesson 1.2 Transformations of Linear and Absolute Value Functions – Page (12-18)
  • Transformations of Linear and Absolute Value Functions 1.2 Exercises – Page (16-18)
  • Linear Functions Study Skills Taking Control of Your Class Time – Page 19
  • Linear Functions 1.1-1.2 Quiz – Page 20
  • Lesson 1.3 Modeling with Linear Functions – Page (22-28)
  • Modeling with Linear Functions 1.3 Exercises – Page (26-28)
  • Lesson 1.4 Solving Linear Systems – Page (30-36)
  • Solving Linear Systems 1.4 Exercises – Page (34-36)
  • Linear Functions Performance Task: Secret of the Hanging Baskets – Page 37
  • Linear Functions Chapter Review – Page (38-40)
  • Linear Functions Chapter Test – Page 41
  • Linear Functions Cumulative Assessment – Page (42-43)

Linear Functions Maintaining Mathematical Proficiency

Evaluate.

Question 1.
5 • 23 + 7

5 • 23 + 7
5.8+7
40+7
47

Question 2.
4 – 2(3 + 2)2

4 – 2(3 + 2)2
4 – 2 (25)
4 – 2 (25)
4 – 50
= – 46

Question 3.
48 ÷ 42 + \(\frac{3}{5}\)
Answer:
48 ÷ 42 + \(\frac{3}{5}\)
48 ÷ 16 + \(\frac{3}{5}\)
3 + \(\frac{3}{5}\) = 3.6

Question 4.
50 ÷ 52 • 2

50 ÷ 52 • 2
50 ÷ 25 . 2
2 .2
4

Question 5.
\(\frac{1}{2}\)(22+ 22)
Answer:
\(\frac{1}{2}\)(22+ 22)
\(\frac{1}{2}\)(4+ 22)
\(\frac{1}{2}\)(26) = 13

Question 6.
\(\frac{1}{6}\)(6 + 18) – 22

Answer:
\(\frac{1}{6}\)(6 + 18) – 22
\(\frac{1}{6}\)(24) – 4
4 – 4 = 0

Graph the transformation of the figure.

Question 7.
Translate the rectangle 1 unit right and 4 units up.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (1)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (2)

Question 8.
Reflect the triangle in the y-axis. Then translate 2 units left.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (3)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (4)

Question 9.
Translate the trapezoid 3 units down. Then reflect in the x-axis.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (5)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (6)

Question 10.
ABSTRACT REASONING Give an example to show why the order of operations is important when evaluating a numerical expression. Is the order of transformations of figures important? Justify your answer.

Theorder of operationstells us theorderto solve steps inexpressionswith more than oneoperation. First, we solve anyoperations inside of parentheses or brackets.

Linear Functions Maintaining Mathematical Practices

Monitoring Progress

Use a graphing calculator to graph the equation using the standard viewing window and a square viewing window. Describe any differences in the graphs.

Question 1.
y = 2x – 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (7)

Question 2.
y = | x + 2 |
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (8)

Question 3.
y = -x2 + 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (9)

Question 4.
y = \(\sqrt{x-1}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (10)

Question 5.
y = x3 – 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (11)

Question 6.
y = 0.25x3

Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (12)

Determine whether the viewing window is square. Explain.

Question 7.
-8 ≤ x ≤ 8, -2 ≤ y ≤ 8
Answer:
-8 ≤ x ≤ 8, -2 ≤ y ≤ 8
The total range of X-axis is 16 units and the total range of Y-axis is 10 units
The ratio of the height to width of the viewing screen is 10/16 = 5/8
So, the ratio is 5:8.
Thus the viewing window is square.

Question 8.
-7 ≤ x ≤ 8, -2 ≤ y ≤ 8
Answer:
-7 ≤ x ≤ 8, -2 ≤ y ≤ 8
The total range of X-axis is 15 units and the total range of Y-axis is 10 units
The ratio of the height to width of the viewing screen is 10/15 = 2/3
So, the ratio is 2:3.
Thus the viewing window is square.

Question 9.
-6 ≤ x ≤ 9, -2 ≤ y ≤ 8
Answer:
-6 ≤ x ≤ 9, -2 ≤ y ≤ 8
The total range of X-axis is 15 units and the total range of Y-axis is 10 units
The ratio of the height to width of the viewing screen is 10/15 = 2/3
So, the ratio is 2:3.
Thus the viewing window is square.

Question 10.
-2 ≤ x≤ 2, -3 ≤ y ≤ 3
Answer:
-2 ≤ x≤ 2, -3 ≤ y ≤ 3
The total range of X-axis is 4 units and the total range of Y-axis is 6 units
The ratio of the height to width of the viewing screen is 6/4 = 3/2
So, the ratio is 3:2.
Thus the viewing window is not a square.

Question 11.
-4 ≤ x ≤ 5, -3 ≤ y ≤ 3
Answer:
-4 ≤ x ≤ 5, -3 ≤ y ≤ 3
The total range of X-axis is 9 units and the total range of Y-axis is 6 units
The ratio of the height to width of the viewing screen is 6/9 = 2/3
So, the ratio is 2:3.
Thus the viewing window is a square.

Question 12.
-4 ≤ x ≤ 4, -3 ≤ y ≤ 3
Answer:
-4 ≤ x ≤ 4, -3 ≤ y ≤ 3
The total range of X-axis is 8 units and the total range of Y-axis is 6 units
The ratio of the height to width of the viewing screen is 6/8 = 2/3
So, the ratio is 2:3.
Thus the viewing window is a square.

Lesson 1.1 Parent Functions and Transformations

Essential Question

What are the characteristics of some of the basic parent functions?
Odd. End behavior go in different directions. If afunctionis positive, the left side of the graph will point down and the right side will point up (increasing from left to right).

EXPLORATION 1
Identifying Basic Parent Functions
Work with a partner.
Graphs of eight basic parent functions are shown below. Classify each function as constant, linear, absolute value, quadratic, square root, cubic, reciprocal, or exponential. Justify your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (13)

Communicate Your Answer

Question 2.
What are the characteristics of some of the basic parent functions?
Key common points of linear parent functions include the fact that the: Equation is y = x. Domain andrangeare real numbers. Slope, or rate of change, is constant.

Question 3.
Write an equation for each function whose graph is shown in Exploration 1. Then use a graphing calculator to verify that your equations are correct.
Answer:

a. The equation for the given graph of absolute value function in exploration 1 is y = |x|
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (14)
b. y = √x
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (15)
c. y = c
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (16)
y = e^x
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (17)
y = x³
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (18)
y = x
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (19)
y = 1/x
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (20)
y = x²
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (21)

1.1 Lesson

Monitoring Progress

Question 1.
Identify the function family to which g belongs. Compare the graph of g to the graph of its parent function.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (22)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (23)

Graph the function and its parent function. Then describe the transformation.

Question 2.
g(x) = x + 3
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (24)

Question 3.
h(x) = (x – 2)2

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (25)

Question 4.
n(x) = – | x |
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (26)

Graph the function and its parent function. Then describe the transformation.

Question 5.
g(x) = 3x
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (27)

Question 6.
h(x) = \(\frac{3}{2}\)x2

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (28)

Question 7.
c(x) = 0.2|x|

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (29)

Use a graphing calculator to graph the function and its parent function. Then describe the transformations

Question 8.
h(x) = –\(\frac{1}{4}\)x + 5

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (30)

Question 9.
d(x) = 3(x – 5)2 – 1
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (31)

Question 10.
The table shows the amount of fuel in a chainsaw over time. What type of function can you use to model the data? When will the tank be empty?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (32)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (33)

Parent Functions and Transformations 1.1 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
The function f(x) = x2 is the ______ of f(x) = 2x2 – 3.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (34)

Question 2.
DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (35)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (36)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (37)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (38)

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, identify the function family to which f belongs. Compare the graph of f to the graph of its parent function.

Question 3.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (39)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (40)

Question 4.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (41)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (42)

Question 5.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (43)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (44)

Question 6.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (45)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (46)

Question 7.
MODELING WITH MATHEMATICS
At 8:00 A.M., the temperature is 43°F. The temperature increases 2°F each hour for the next 7 hours. Graph the temperatures over time t (t = 0 represents 8:00 A.M.). What type of function can you use to model the data? Explain.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (47)

Question 8.
MODELING WITH MATHEMATICS
You purchase a car from a dealership for $10,000. The trade-in value of the car each year after the purchase is given by the function f(x) = 10,000 – 250x2. What type of function models the trade-in value?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (48)

In Exercises 9–18, graph the function and its parent function. Then describe the transformation.

Question 9.
g(x) = x + 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (49)

Question 10.
f(x) = x – 6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (50)

Question 11.
f(x) = x2 – 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (51)

Question 12.
h(x) = (x+ 4)2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (52)

Question 13.
g(x) = | x – 5 |
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (53)

Question 14.
f(x) = 4 + | x |
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (54)

Question 15.
h(x) = -x2
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (55)
Question 16.
g(x) = -x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (56)

Question 17.
f(x) = 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (57)

Question 18.
f(x) = -2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (58)

In Exercises 19–26, graph the function and its parent function. Then describe the transformation.

Question 19.
f(x) = \(\frac{1}{3}\)x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (59)

Question 20.
g(x) = 4x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (60)

Question 21.
f(x) = 2x2

Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (61)

Question 22.
h(x) = \(\frac{1}{3}\)x2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (62)

Question 23.
h(x) = \(\frac{3}{4}\)x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (63)

Question 24.
g(x) = \(\frac{4}{3}\)x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (64)

Question 25.
h(x) = 3 | x |
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (65)

Question 26.
f(x) = \(\frac{1}{2}\) | x |
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (66)

In Exercises 27–34, use a graphing calculator to graph the function and its parent function. Then describe the transformations.

Question 27.
f(x) = 3x + 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (67)

Question 28.
h(x) = -x + 5
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (68)

Question 29.
h(x) = -3 | x | – 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (69)

Question 30.
f(x) = \(\frac{3}{4}\) | x | + 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (70)

Question 31.
g(x) = \(\frac{1}{2}\)x2 – 6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (71)

Question 32.
f(x) = 4x2 – 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (72)

Question 33.
f(x) = -(x + 3)2 + \(\frac{1}{4}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (73)

Question 34.
g(x) = – | x – 1 | – \(\frac{1}{2}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (74)

ERROR ANALYSIS In Exercises 35 and 36, identify and correct the error in describing the transformation of the parent function.

Question 35.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (75)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (76)

Question 36.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (77)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (78)

MATHEMATICAL CONNECTIONS In Exercises 37 and 38, find the coordinates of the figure after the transformation.

Question 37.
Translate 2 units down.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (79)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (80)

Question 38.
Reflect in the x-axis.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (81)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (82)

USING TOOLS In Exercises 39–44, identify the function family and describe the domain and range. Use a graphing calculator to verify your answer.

Question 39.
g(x) = | x + 2 | – 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (83)

Question 40.
h(x) = | x – 3 | + 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (84)

Question 41.
g(x) = 3x + 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (85)

Question 42.
f(x) = -4x + 11
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (86)

Question 43.
f(x) = 5x2 – 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (87)

Question 44.
f(x) = -2x2 + 6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (88)

Question 45.
MODELING WITH MATHEMATICS The table shows the speeds of a car as it travels through an intersection with a stop sign. What type of function can you use to model the data? Estimate the speed of the car when it is 20 yards past the intersection.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (89)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (90)

Question 46.
THOUGHT PROVOKING In the same coordinate plane, sketch the graph of the parent quadratic function and the graph of a quadratic function that has no x-intercepts. Describe the transformation(s) of the parent function.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (91)

Question 47.
USING STRUCTURE Graph the functions f(x) = | x – 4 | and g(x) = | x | – 4. Are they equivalent? Explain.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (92)

Question 48.
HOW DO YOU SEE IT? Consider the graphs of f, g, and h.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (93)
a. Does the graph of g represent a vertical stretch or a vertical shrink of the graph of f? Explain your reasoning.
b. Describe how to transform the graph of f to obtain the graph of h.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (94)

Question 49.
MAKING AN ARGUMENT Your friend says two different translations of the graph of the parent linear function can result in the graph of f(x) = x – 2. Is your friend correct? Explain.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (95)

Question 50.
DRAWING CONCLUSIONS A person swims at a constant speed of 1 meter per second. What type of function can be used to model the distance the swimmer travels? If the person has a 10-meter head start, what type of transformation does this represent? Explain.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (96)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (97)

Question 51.
PROBLEM SOLVING You are playing basketball with your friends. The height (in feet) of the ball above the ground t seconds after a shot is released from your hand is modeled by the function f(t) = -16t2 + 32t + 5.2.
a. Without graphing, identify the type of function that models the height of the basketball.
b. What is the value of t when the ball is released from your hand? Explain your reasoning.
c. How many feet above the ground is the ball when it is released from your hand? Explain.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (98)

Question 52.
MODELING WITH MATHEMATICS The table shows the battery lives of a computer over time. What type of function can you use to model the data? Interpret the meaning of the x-intercept in this situation.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (99)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (100)

Question 53.
REASONING Compare each function with its parent function. State whether it contains a horizontal translation, vertical translation, both, or neither. Explain your reasoning.
a. f(x) = 2 | x | – 3
b. f(x) = (x – 8)2
c. f(x) = | x + 2 | + 4
d. f(x) = 4x2

Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (101)

Question 54.
CRITICAL THINKING
Use the values -1, 0, 1, and 2 in the correct box so the graph of each function intersects the x-axis. Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (102)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (103)

Maintaining Mathematical Proficiency

Determine whether the ordered pair is a solution of the equation. (Skills Review Handbook)

Question 55.
f(x) = | x + 2 |; (1, -3)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (104)

Question 56.
f(x) = | x | – 3; (-2, -5)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (105)

Question 57.
f(x) = x – 3; (5, 2)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (106)

Question 58.
f(x) = x – 4; (12, 8)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (107)

Find the x-intercept and the y-intercept of the graph of the equation. (Skills Review Handbook)

Question 59.
y = x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (108)

Question 60.
y = x + 2
Answer:
To find the x-intercept let y = 0, the solve for x.
y = x + 2
0 = x + 2
x = -2
To find the y-intercept let x = 0, then solve for y.
y = x+ 2
y = 0 + 2
y = 2
So, the intercept is (0, 0) and the y-intercept is (-2, 2)

Question 61.
3x + y = 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (109)

Question 62.
x – 2y = 8
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (110)

Lesson 1.2 Transformations of Linear and Absolute Value Functions

Essential Question

How do the graphs of y = f(x) + k, y = f(x – h), and y = -f(x) compare to the graph of the parent function f?

EXPLORATION 1
Transformations of the Parent Absolute Value Function
Work with a partner.
Compare the graph of the function
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (111)
y = | x | + k Transformation
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (112)
to the graph of the parent function
f(x) = | x |.

EXPLORATION 2
Transformations of the Parent Absolute Value Function
Work with a partner.
Compare the graph of the function
y = | x – h | Transformation
to the graph of the parent function
f(x) = | x |. Parent function
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (113)

EXPLORATION 3
Transformation of the Parent Absolute Value Function
Work with a partner.
Compare the graph of the function
y = – | x | Transformation
to the graph of the parent function
f(x) = | x | Parent function
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (114)

Communicate Your Answer

Question 4.
Transformation How do the graphs of y = f (x) + k, y = f(x – h), and y = -f(x) compare to the graph of the parent function f?

Answer:
The graphs of y = f (x) + k, y = f(x – h), and y = -f(x) are compared to the graph of the parent function by vertical shifts, horizontal shifts and reflections.

  • Vertical shifts: Let f(x) be the parent function and k be a positive number. To graph the function y = f (x) + k, we shift the graph of y = f(x) up k units by adding k to the y-coordinates of the points on the graph of f.
  • Horizontal shifts: Let f(x) be the parent function and h be a positive number. To graph the function y = f(x – h), we shift the graph of y = f(x) to the right h units by adding h to the x-coordinates of the points on the graph of f.
  • Reflections: Let f(x) be the parent function.
    To graph the function y = -f(x), we reflect the graph of y = f(x) about the x-axis by multiplying the y-coordinates of the points on the graph of f by -1.

Question 5.
Compare the graph of each function to the graph of its parent function f. Use a graphing calculator to verify your answers are correct.
a. y = \([\sqrt{x}/latex] – 4
b. y = [latex][\sqrt{x + 4}/latex]
c. y = –[latex][\sqrt{x}/latex]
d. y = x2 + 1
e. y = (x – 1)2
f. y = -x2

1.2 Lesson

Monitoring Progress

Write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer.

Question 1.
f(x) = 3x; translation 5 units up
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (115)

Question 2.
f(x) = | x | – 3; translation 4 units to the right
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (116)

Question 3.
f(x) = – | x + 2 | – 1; reflection in the x-axis
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (117)

Question 4.
f(x) = [latex]\frac{1}{2}\)x+ 1; reflection in the y-axis
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (118)

Write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer.

Question 5.
f(x) = 4x+ 2; horizontal stretch by a factor of 2

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (119)

Question 6.
f(x) = | x | – 3; vertical shrink by a factor of \(\frac{1}{3}\)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (120)

Question 7.
Let the graph of g be a translation 6 units down followed by a reflection in the x-axis of the graph of f(x) = | x |. Write a rule for g. Use a graphing calculator to check your answer.

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (121)

Question 8.
WHAT IF? In Example 5, your revenue function is f(x) = 3x. How does this affect your profit for 100 downloads?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (122)

Transformations of Linear and Absolute Value Functions 1.2 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
The function g(x) = | 5x |- 4 is a horizontal ___________ of the function f(x) = | x | – 4.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (123)

Question 2.
WHICH ONE DOESN’T BELONG? Which transformation does not belong with the other three? Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (124)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (125)

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer.

Question 3.
f(x) = x – 5; translation 4 units to the left
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (126)

Question 4.
f(x) = x + 2; translation 2 units to the right
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (127)

Question 5.
f(x) = | 4x + 3 | + 2; translation 2 units down
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (128)

Question 6.
f(x) = 2x – 9; translation 6 units up
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (129)

Question 7.
f(x) = 4 – | x + 1 |
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (130)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (131)

Question 8.
f(x) = | 4x | + 5
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (132)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (133)

Question 9.
WRITING Describe two different translations of the graph of f that result in the graph of g.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (134)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (135)

Question 10.
PROBLEM SOLVING You open a café. The function f(x) = 4000x represents your expected net income (in dollars) after being open x weeks. Before you open, you incur an extra expense of $12,000. What transformation of f is necessary to model this situation? How many weeks will it take to pay off the extra expense?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (136)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (137)

In Exercises 11–16, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer.

Question 11.
f(x) = -5x+ 2; reflection in the x-axis
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (138)

Question 12.
f(x) = \(\frac{1}{2}\)x – 3; reflection in the x-axis
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (139)

Question 13.
f(x) = | 6x | – 2; reflection in the y-axis
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (140)

Question 14.
f(x) = | 2x – 1 | + 3; reflection in the y-axis
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (141)

Question 15.
f(x) = -3 + | x – 11 |; reflection in the y-axis
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (142)

Question 16.
f(x) = -x+ 1; reflection in the y-axis
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (143)

In Exercises 17–22, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer.

Question 17.
f(x) = x + 2; vertical stretch by a factor of 5
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (144)

Question 18.
f(x) = 2x+ 6; vertical shrink by a factor of \(\frac{1}{2}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (145)

Question 19.
f(x) = | 2x | + 4; horizontal shrink by a factor of \(\frac{1}{2}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (146)

Question 20.
f(x) = | x+ 3 | ; horizontal stretch by a factor of 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (147)

Question 21.
f(x) = -2 | x – 4 | + 2
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (148)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (149)

Question 22.
f(x) = 6 – x
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (150)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (151)

ANALYZING RELATIONSHIPS In Exercises 23–26, match the graph of the transformation of f with the correct equation shown. Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (152)

Question 23.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (153)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (154)

Question 24.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (155)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (156)

Question 25.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (157)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (158)

Question 26.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (159)
Answer:
A. y = 2f(x)
B. y = f(2x)
C. y = f(x + 2)
D. y = f(x) + 2

In Exercises 27–32, write a function g whose graph represents the indicated transformations of the graph of f.

Question 27.
f(x) = x; vertical stretch by a factor of 2 followed by a translation 1 unit up
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (160)

Question 28.
f(x) = x; translation 3 units down followed by a vertical shrink by a factor of \(\frac{1}{3}\)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (161)

Question 29.
f(x) = | x | ; translation 2 units to the right followed by a horizontal stretch by a factor of 2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (162)

Question 30.
f(x) = | x |; reflection in the y-axis followed by a translation 3 units to the right
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (163)

Question 31.
f(x) = | x |
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (164)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (165)

Question 32.
f(x) = | x |
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (166)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (167)

ERROR ANALYSIS In Exercises 33 and 34, identify and correct the error in writing the function g whose graph represents the indicated transformations of the graph of f.

Question 33.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (168)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (169)

Question 34.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (170)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (171)

Question 35.
MAKING AN ARGUMENT Your friend claims that when writing a function whose graph represents a combination of transformations, the order is not important. Is your friend correct? Justify your answer.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (172)

Question 36.
MODELING WITH MATHEMATICS During a recent period of time, bookstore sales have been declining. The sales (in billions of dollars) can be modeled by the function f(t) = –\(\frac{7}{5}\)t + 17.2, where t is the number of years since 2006. Suppose sales decreased at twice the rate. How can you transform the graph of f to model the sales? Explain how the sales in 2010 are affected by this change.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (173)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (174)

MATHEMATICAL CONNECTIONS For Exercises 37–40, describe the transformation of the graph of f to the graph of g. Then find the area of the shaded triangle.

Question 37.
f(x) = | x – 3 |
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (175)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (176)

Question 38.
f(x) = – | x | – 2
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (177)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (178)

Question 39.
f(x) = -x + 4
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (179)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (180)

Question 40.
f(x) = x – 5
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (181)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (182)

Question 41.
ABSTRACT REASONING The functions f(x) = mx + b and g(x) = mx + c represent two parallel lines.
a. Write an expression for the vertical translation of the graph of f to the graph of g.
b. Use the definition of slope to write an expression for the horizontal translation of the graph of f to the graph of g.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (183)

Question 42.
HOW DO YOU SEE IT? Consider the graph of f(x) = mx + b. Describe the effect each transformation has on the slope of the line and the intercepts of the graph.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (184)
a. Reflect the graph of f in the y-axis.
b. Shrink the graph of f vertically by a factor of \(\frac{1}{3}\).
c. Stretch the graph of f horizontally by a factor of 2.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (185)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (186)

Question 43.
REASONING The graph of g(x) = -4 |x | + 2 is a reflection in the x-axis, vertical stretch by a factor of 4, and a translation 2 units down of the graph of its parent function. Choose the correct order for the transformations of the graph of the parent function to obtain the graph of g. Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (187)

Question 44.
THOUGHT PROVOKING You are planning a cross-country bicycle trip of 4320 miles. Your distance d (in miles) from the halfway point can be modeled by d = 72 |x – 30 |, where x is the time (in days) and x = 0 represents June 1. Your plans are altered so that the model is now a right shift of the original model. Give an example of how this can happen. Sketch both the original model and the shifted model.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (188)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (189)

Question 45.
CRITICAL THINKING Use the correct value 0, -2, or 1 with a, b, and c so the graph of g(x) = a|x – b | + c is a reflection in the x-axis followed by a translation one unit to the left and one unit up of the graph of f(x) = 2 |x – 2 | + 1. Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (190)

Maintaining Mathematical Proficiency

Evaluate the function for the given value of x. (Skills Review Handbook)

Question 46.
f(x) = x + 4; x = 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (191)

Question 47.
f(x) = 4x – 1; x = -1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (192)

Question 48.
f(x) = -x + 3; x = 5
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (193)

Question 49.
f(x) = -2x – 2; x = -1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (194)

Create a scatter plot of the data. (Skills Review Handbook)

Question 50.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (195)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (196)

Question 51.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (197)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (198)

Linear Functions Study Skills Taking Control of Your Class Time

1.1 – 1.2 What Did You Learn?

Core Vocabulary
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (199)

Core Concepts

Section 1.1
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (200)

Section 1.2
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (201)

Mathematical Practices

Question 1.
How can you analyze the values given in the table in Exercise 45 on page 9 to help you determine what type of function models the data?

Question 2.
Explain how you would round your answer in Exercise 10 on page 16 if the extra expense is $13,500.

Study Skills

Taking Control of Your Class Time

Question 1.
Sit where you can easily see and hear the teacher, and the teacher can see you.

Question 2.
Pay attention to what the teacher says about math, not just what is written on the board.

Question 3.
Ask a question if the teacher is moving through the material too fast.

Question 4.
Try to memorize new information while learning it.

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (202)

Question 5.
Ask for clarification if you do not understand something.

Question 6.
Think as intensely as if you were going to take a quiz on the material at the end of class.

Question 7.
Volunteer when the teacher asks for someone to go up to the board.

Question 8.
At the end of class, identify concepts or problems for which you still need clarification.

Question 9.
Use the tutorials at BigIdeasMath.com for additional help.

Linear Functions 1.1-1.2 Quiz

Identify the function family to which g belongs. Compare the graph of the function to the graph of its parent function. (Section 1.1)

Question 1.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (203)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (204)

Question 2.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (205)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (206)

Question 3.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (207)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (208)

Graph the function and its parent function. Then describe the transformation. (Section 1.1)

Question 4.
f(x) = \(\frac{3}{2}\)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (209)

Question 5.
f(x) = 3x
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (210)

Question 6.
f(x) = 2(x – 1)2

Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (211)

Question 7.
f(x) = – | x + 2 | – 7
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (212)

Question 8.
f(x) = \(\frac{1}{4}\)x2 + 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (213)

Question 9.
f(x) = –\(\frac{1}{2}\)x – 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (214)

Write a function g whose graph represents the indicated transformation of the graph of f. (Section 1.2)

Question 10.
f(x) = 2x + 1; translation 3 units up
Answer:
The vertex is (0, 1) of the original graph to move the vertex up by 3 units just add 3 to the y-intercept.
g(x) = 2x + 4

Question 11.
f(x) = -3 | x – 4 | ; vertical shrink by a factor of \(\frac{1}{2}\)
Answer:
f(x) = -3 | x – 4 | to vertically shrink a function by a factor by c, multiply the whole function by c f(x) vertically shrunk by a factor of c would be cf(x)
so f(x) = -3 | x – 4 | vertically shrunk by a factor of 1/2 would be f(x) = (-3/2) |x – 4|

Question 12.
f(x) = 3 | x + 5 |; reflection in the x-axis
Answer:
f(x) = 3 | x + 5 |
The points reflected in the x-axis have opposite y-coordinates
f(x) = -y
-y = -3 |x + 5|
f(x) = -3 |x + 5|

Question 13.
f(x) = \(\frac{1}{3}\)x – \(\frac{2}{3}\) ; translation 4 units left
Answer:
f(x) = \(\frac{1}{3}\)x – \(\frac{2}{3}\)
f(x) = \(\frac{1}{3}\) (x – 2)
f(x) = \(\frac{1}{3}\) (x – 2 + 4)
f(x) = \(\frac{1}{3}\) (x + 2)
f(x) = \(\frac{1}{3}\)x + \(\frac{2}{3}\)

Write a function g whose graph represents the indicated transformations of the graph of f. (Section 1.2)

Question 14.
Let g be a translation 2 units down and a horizontal shrink by a factor of \(\frac{2}{3}\) of the graph of f(x) =x.
Answer:
f(x) =x
horizontal shrink by a factor of \(\frac{2}{3}\)
f(x) = \(\frac{3}{2}\)x
f(x) = \(\frac{3}{2}\)x – 2
g(x) =\(\frac{3}{2}\)x – 2

Question 15.
Let g be a translation 9 units down followed by a reflection in the y-axis of the graph of f(x) = x.
Answer:
f(x) =x
g(x) = f(x) – 9
It is also reflected about the y-axis.
g(x) = f(-x) – 9

Question 16.
Let g be a reflection in the x-axis and a vertical stretch by a factor of 4 followed by a translation 7 units down and 1 unit right of the graph of f(x) = | x |.
Answer:
f(x) = |x|
g(x) = bf(x)
Reflecting function over the x-axis
g(x) = -f(x)
Original function is f(x) = -|x|
Stretching by a factor of 4 means we have to multiply by 4.
g(x) = -4|x|
g(x) = -4x
translation 7 units down and 1 unit right of the graph of f(x) = | x |
g(x) = -4x – 7 -1
g(x) = -4x – 8
g(x) = -4(x + 2)

Question 17.
Let g be a translation 1 unit down and 2 units left followed by a vertical shrink by a factor of \(\frac{1}{2}\) of the graph of f(x) = | x |.
Answer:
f(x) = |x|
Multiply output with 1/2 to vertically shrink function
g(x) = 1/2 |x|
Subtract 1 to output of function to translate 1 unit down and 2 units left.
g(x) = 1/2 |x| – 1 -2
g(x) = 1/2 |x| – 3

Question 18.
The table shows the total distance a new car travels each month after it is purchased. What type of function can you use to model the data? Estimate the mileage after 1 year. (Section 1.1)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (215)
Answer:
From the given data in the above table,
2 months = 2300
1 month = 2300/2 = 1150
12 months = 1150 × 12 = 13800
The mileage after 1 year is 13800.
The estimated mileage after 1 year is 14,000 miles.

Question 19.
The total cost of an annual pass plus camping for x days in a National Park can be modeled by the function f(x) = 20x+ 80. Senior citizens pay half of this price and receive an additional $30 discount. Describe how to transform the graph of f to model the total cost for a senior citizen. What is the total cost for a senior citizen to go camping for three days? (Section 1.2)
Answer:
Total cost of annual pass plus camping for x days f(x) = 2x + 80
For senior citizen cost
= 1/2 (20x + 80) – 30
= 10x + 40 – 30
= 10x + 10
= 10(x + 1)
x = 3
= 10(3 + 1)
= 10(4)
= $40

Lesson 1.3 Modeling with Linear Functions

Essential Question
How can you use a linear function to model and analyze a real-life situation?

EXPLORATION 1
Modeling with a Linear Function
Work with a partner.
A company purchases a copier for $12,000. The spreadsheet shows how the copier depreciates over an 8-year period.
a. Write a linear function to represent the value V of the copier as a function of the number t of years.
b. Sketch a graph of the function. Explain why this type of depreciation is called straight line depreciation.
c. Interpret the slope of the graph in the context of the problem.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (216)
Answer:
m = 12,000 – 10,750 = $1,250
b = 12,000
V = -1250t + 12,000
The slope shows that for every year that passes, the value depreciates by $1250

EXPLORATION 2
Modeling with Linear Functions
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (217)
Work with a partner. Match each description of the situation with its corresponding graph. Explain your reasoning.
a. A person gives $20 per week to a friend to repay a $200 loan.
b. An employee receives $12.50 per hour plus $2 for each unit produced per hour.
c. A sales representative receives $30 per day for food plus $0.565 for each mile driven.
d. A computer that was purchased for $750 depreciates $100 per year.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (218)

Communicate Your Answer

Question 3.
How can you use a linear function to model and analyze a real-life situation?
Answer:
One of the real life situation is finding Variable costs.
Imagine that you are taking a taxi while on vacation. You know that the taxi service charges 9 rupees to pick your family up from your hotel and another 0.15 rupees per mile for the trip. Without knowing how many miles it will be to each destination, you can set up a linear equation that can be used to find the cost of any taxi trip you take on your trip. By using′ ′x′′ to represent the number of miles to your destination and ”y′′ to represent the cost of that taxi ride, the linear equation would be: y = 0.15x+9
For Example: If one company offers to pay you 450 rupees per week and the other offers 10 rupees per hour, and both ask you to work 40 hours per week, which company is offering the better rate of pay? A linear equation can help you figure it out.
The first company’s offer is expressed as 450 = 40x.
The second company’s offer is expressed as y = 10.
After comparing the two offers, the equations tell you the first company is offering a better rate of pay at 11.25 rupees per hour.

Question 4.
Use the Internet or some other reference to find a real-life example of straight line depreciation.
a. Use a spreadsheet to show the depreciation.
b. Write a function that models the depreciation.
c. Sketch a graph of the function.
Answer:
The real life example of straight line depreciation is the decrease of speed of car by ten meter per second which was moving with an initial speed of a hundred meter per second till speed reaches thirty meter per second.

SpeedTime (in second)
1000
901
802
703
604
505
406
307

b. The real life example of straight line depreciation is the decrease of speed of car by ten meter per second which was moving with an initial speed of a hundred meter per second till speed reaches thirty meter per second.
The function that represents the statement is y = 100 – 10x, where y is speed and x is time.
c.
y = 100 – 10x
x = 7
y = 100 – 10(7)
y = 100 – 70
y = 30
when x = 0
y = 100
when x = 4
y = 60
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (219)

1.3 Lesson

Monitoring Progress

Question 1.
The graph shows the remaining balance y on a car loan after making x monthly payments. Write an equation of the line and interpret the slope and y-intercept. What is the remaining balance after 36 payments?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (220)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (221)
Question 2.
WHAT IF? Maple Ridge charges a rental fee plus a $10 fee per student. The total cost is $1900 for 140 students. Describe the number of students that must attend for the total cost at Maple Ridge to be less than the total costs at the other two venues. Use a graph to justify your answer.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (222)

Question 3.
The table shows the humerus lengths (in centimeters) and heights (in centimeters) of several females.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (223)
a. Do the data show a linear relationship? If so, write an equation of a line of fit and use it to estimate the height of a female whose humerus is 40 centimeters long.
b. Use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data. Estimate the height of a female whose humerus is 40 centimeters long. Compare this height to your estimate in part (a).
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (224)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (225)

Modeling with Linear Functions 1.3 Exercises

Question 1.
COMPLETE THE SENTENCE The linear equation y = \(\frac{1}{2}\)x + 3 is written in ____________ form.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (226)

Question 2.
VOCABULARY A line of best fit has a correlation coefficient of -0.98. What can you conclude about the slope of the line?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (227)

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, use the graph to write an equation of the line and interpret the slope.

Question 3.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (228)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (229)

Question 4.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (230)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (231)

Question 5.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (232)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (233)

Question 6.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (234)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (235)

Question 7.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (236)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (237)

Question 8.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (238)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (239)

Question 9.
MODELING WITH MATHEMATICS Two newspapers charge a fee for placing an advertisem*nt in their paper plus a fee based on the number of lines in the advertisem*nt. The table shows the total costs for different length advertisem*nts at the Daily Times. The total cost y (in dollars) for an advertisem*nt that is x lines long at the Greenville Journal is represented by the equation y = 2x + 20. Which newspaper charges less per line? How many lines must be in an advertisem*nt for the total costs to be the same?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (240)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (241)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (242)

Question 10.
PROBLEM SOLVING While on vacation in Canada, you notice that temperatures are reported in degrees Celsius. You know there is a linear relationship between Fahrenheit and Celsius, but you forget the formula. From science class, you remember the freezing point of water is 0°C or 32°F, and its boiling point is 100°C or 212°F.
a. Write an equation that represents degrees Fahrenheit in terms of degrees Celsius.
b. The temperature outside is 22°C. What is this temperature in degrees Fahrenheit?
c. Rewrite your equation in part (a) to represent degrees Celsius in terms of degrees Fahrenheit.
d. The temperature of the hotel pool water is 83°F. What is this temperature in degrees Celsius?
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (243)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (244)

ERROR ANALYSIS In Exercises 11 and 12, describe and correct the error in interpreting the slope in the context of the situation.

Question 11.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (245)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (246)

Question 12.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (247)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (248)

In Exercises 13–16, determine whether the data show a linear relationship. If so, write an equation of a line of fit. Estimate y when x = 15 and explain its meaning in the context of the situation.

Question 13.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (249)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (250)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (251)

Question 14.

Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (252)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (253)

Question 15.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (254)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (255)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (256)

Question 16.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (257)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (258)

Question 17.
MODELING WITH MATHEMATICS The data pairs (x, y) represent the average annual tuition y (in dollars) for public colleges in the United States x years after 2005. Use the linear regression feature on a graphing calculator to find an equation of the line of best fit. Estimate the average annual tuition in 2020. Interpret the slope and y-intercept in this situation.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (259)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (260)

Question 18.
MODELING WITH MATHEMATICS The table shows the numbers of tickets sold for a concert when different prices are charged. Write an equation of a line of fit for the data. Does it seem reasonable to use your model to predict the number of tickets sold when the ticket price is $85? Explain.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (261)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (262)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (263)

USING TOOLS In Exercises 19–24, use the linear regression feature on a graphing calculator to find an equation of the line of best fit for the data. Find and interpret the correlation coefficient.

Question 19.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (264)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (265)

Question 20.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (266)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (267)

Question 21.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (268)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (269)

Question 22.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (270)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (271)

Question 23.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (272)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (273)

Question 24.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (274)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (275)

Question 25.
OPEN-ENDED Give two real-life quantities that have
(a) a positive correlation,
(b) a negative correlation, and
(c) approximately no correlation. Explain.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (276)

Question 26.
HOW DO YOU SEE IT? You secure an interest-free loan to purchase a boat. You agree to make equal monthly payments for the next two years. The graph shows the amount of money you still owe.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (277)
a. What is the slope of the line? What does the slope represent?
Answer:
According to the graph (0, 30), (24, 0) are on the line.
We can assume the slope is k.
k = 30-0/0-24 = -5/4
And the slope represents the amount of money to be paid monthly.

b. What is the domain and range of the function? What does each represent?
Answer:
Because x and y are linear function, and k = -5/4, (0, 30) is on the function.
y = -5/4 x + 30
And the domain is [0, 24] and the range is [0, 30]
and the range represents the total money still to be paid.

c. How much do you still owe after making payments for 12 months?
Answer:
Because y = -5/4 x + 30
when x = 12, y = 15
So, you still owe 15 hundreds of dollars.

Question 27.
MAKING AN ARGUMENT A set of data pairs has a correlation coefficient r = 0.3. Your friend says that because the correlation coefficient is positive, it is logical to use the line of best fit to make predictions. Is your friend correct? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (278)
Question 28.
THOUGHT PROVOKING Points A and B lie on the line y = -x + 4. Choose coordinates for points A, B, and C where point C is the same distance from point A as it is from point B. Write equations for the lines connecting points A and C and points B and C.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (279)

Question 29.
ABSTRACT REASONING If x and y have a positive correlation, and y and z have a negative correlation, then what can you conclude about the correlation between x and z? Explain.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (280)

Question 30.
MATHEMATICAL CONNECTIONS Which equation has a graph that is a line passing through the point (8, -5) and is perpendicular to the graph of y = -4x + 1?
A. y = \(\frac{1}{4}\)x – 5
B. y = -4x + 27
C. y = –\(\frac{1}{4}\)x – 7
D. y = \(\frac{1}{4}\)x – 7
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (281)

Question 31.
PROBLEM SOLVING You are participating in an orienteering competition. The diagram shows the position of a river that cuts through the woods. You are currently 2 miles east and 1 mile north of your starting point, the origin. What is the shortest distance you must travel to reach the river?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (282)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (283)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (284)

Question 32.
ANALYZING RELATIONSHIPS Data from North American countries show a positive correlation between the number of personal computers per capita and the average life expectancy in the country.
a. Does a positive correlation make sense in this situation? Explain.
b. Is it reasonable to conclude that giving residents of a country personal computers will lengthen their lives? Explain.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (285)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (286)

Maintaining Mathematical Proficiency

Solve the system of linear equations in two variables by elimination or substitution. (Skills Review Handbook)

Question 33.
3x + y = 7
-2x – y = 9
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (287)

Question 34.
4x + 3y = 2
2x – 3y = 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (288)

Question 35.
2x + 2y = 3
x = 4y – 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (289)

Question 36.
y = 1 + x
2x + y = -2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (290)

Question 37.
\(\frac{1}{2}\)x + 4y = 4
2x – y = 1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (291)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (292)

Question 38.
y = x – 4
4x + y = 26
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (293)

Lesson 1.4 Solving Linear Systems

Essential Question
How can you determine the number of solutions of a linear system?
A linear system is consistent when it has at least one solution. A linear system is inconsistent when it has no solution.

EXPLORATION 1
Recognizing Graphs of Linear Systems
Work with a partner.
Match each linear system with its corresponding graph. Explain your reasoning. Then classify the system as consistent or inconsistent.
a. 2x – 3y = 3
-4x + 6y = 6
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (294)
Answer:
2x – 3y = 3 —- × 2 ⇒ 4x – 6y = 6
-4x + 6y = 6

4x – 6y = 6
-4x + 6y = 6
0
So, the linear system is inconsistent.

b. 2x – 3y = 3
x + 2y = 5
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (295)
Answer:
2x – 3y = 3
x + 2y = 5 —– × 2
2x + 4y = 10

2x – 3y = 3
(-)2x + 4y = 10
-7y = -7
y = 1
x + 2 = 5
x = 3
So, x = 3 and y = 1
So, the system is consistent

c. 2x – 3y = 3
-4x + 6y = 6
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (296)
Answer:
2x – 3y = 3 —- × 2
-4x + 6y = 6

4x – 6y = 6
-4x + 6y = 6
0
So, the linear system is inconsistent.

EXPLORATION 2
Solving Systems of Linear Equations
Work with a partner.
Solve each linear system by substitution or elimination. Then use the graph of the system below to check your solution.
a. 2x + y = 5
x – y = 1
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (297)
Answer:
2x + y = 5
x – y = 1
3x = 6
x = 2
2 – y = 1
2 – 1 = y
y = 1
The linear system has one solution.

b. x+ 3y = 1
-x + 2y = 4
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (298)
Answer:
x+ 3y = 1
-x + 2y = 4
5y = 5
y = 1
x + 3 = 1
x = 1 – 3
x = -2
The linear system has one solution.

c. x + y = 0
3x + 2y = 1
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (299)

Answer:
x + y = 0 — × 2 = 2x + 2y = 0
3x + 2y = 1

2x + 2y = 0
(-)3x + 2y = 1
-x = -1
x = 1
The linear system has one solution.

Communicate Your Answer

Question 3.
How can you determine the number of solutions of a linear system?
Answer:
A linear equation in two variables is an equation of the form ax + by + c = 0 where a, b, c ∈ R, a, and b ≠ 0. When we consider a system of linear equations, we can find the number of solutions by comparing the coefficients of the variables of the equations.

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (300)
Question 4.
Suppose you were given a system of three linear equations in three variables. Explain how you would approach solving such a system.
Answer:

  • Solve one equation for one of its variables.
  • Substitute the expression from point 1 in the other two equations to obtain a linear system in two variables.
  • Solve the new linear system for both of its variables.
  • Substitute the values found in point 3 into one of the original equations and solve for the remaining variable.

Question 5.
Apply your strategy in Question 4 to solve the linear system.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (301)
Answer:
x + y + z = 1
x – y – z = 3
-x – y – z = -1
Solving 1 & 2
x + y + z = 1
x – y – z = 3
2x = 4
x = 4/2
x = 2
x – y – z = 1
-x – y – z = -1
-2y -2z = 0
y + z = 0
y = -z

1.4 Lesson

Monitoring Progress

Question 1.
x – 2y + z = -11
3x + 2y – z = 7
-x + 2y + 4z = -9
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (302)

Question 2.
x + y – z = -1
4x + 4y – 4z = -2
3x + 2y + z = 0
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (303)

Question 3.
x + y + z = 8
x – y + z = 8
2x + y + 2z = 16
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (304)

Question 4.
In Example 3, describe the solutions of the system using an ordered triple in terms of y.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (305)

Question 5.
WHAT IF? On the first day, 10,000 tickets sold, generating $356,000 in revenue. The number of seats sold in Sections A and B are the same. How many lawn seats are still available?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (306)

Solving Linear Systems 1.4 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY The solution of a system of three linear equations is expressed as a(n)__________.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (307)

Question 2.
WRITING Explain how you know when a linear system in three variables has infinitely many solutions.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (308)

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, solve the system using the elimination method.

Question 3.
x + y – 2z = 5
-x + 2y + z = 2
2x + 3y – z = 9
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (309)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (310)

Question 4.
x + 4y – 6z = -1
2x – y + 2z = -7
-x + 2y – 4z = 5
Answer:
x + 4y – 6z = -1— (1)
2x – y + 2z = -7 —- (2)
-x + 2y – 4z = 5 —-(3)
Solving 1 & 3
x + 4y – 6z = -1
-x + 2y – 4z = 5
6y – 10z = 4
3y – 5z = 2 — (4)
Solving (1) & (2)
x + 4y – 6z = -1 — × 2 ⇒ 2x + 6y – 12z = -2
2x – y + 2z = -7

2x + 6y – 12z = -2
2x – y + 2z = -7
– + – +
7y – 14z = 5 —- (5)
Solving (2) & (3)
2x – y + 2z = -7 —- (2)
-x + 2y – 4z = 5 —-(3)—–×2 ⇒ -2x + 4y – 8z = 10

2x – y + 2z = -7
-2x + 4y – 8z = 10
3y – 5z = 3 — (6)
Solving 5 & 6
7y – 14z = 5 — × 3 ⇒ 21y – 42z = 15 — (7)
3y – 5z = 3 — × 7 ⇒ 21y – 35z = 21 — (8)

21y – 42z = 15
21y – 35z = 21
-7z = -6
z = 6/7 or 0.85
7y – 14z = 5
7y – 14(6/7) = 5
7y – 14(6) = 5
7y – 84 = 5
7y = 5 + 84
7y = 89
y = 89/7
y = 12.7

-x + 2y – 4z = 5
-x + 2(12.7) – 4(0.85) = 5
-x + 25.4 – 3.4 = 5
-x + 22 = 5
-x = 5 – 22
-x = -17
x = 17
The solution is (17, 12.7, 0.85)

Question 5.
2x + y – z = 9
-x + 6y + 2z = -17
5x + 7y + z = 4
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (311)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (312)

Question 6.
3x + 2y – z = 8
-3x + 4y + 5z = -14
x – 3y + 4z = -14

Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (313)

Question 7.
2x + 2y + 5z = -1
2x – y + z = 2
2x + 4y – 3z = 14
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (314)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (315)

Question 8.
3x + 2y – 3z = -2
7x – 2y + 5z = -14
2x + 4y + z = 6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (316)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (317)

ERROR ANALYSIS In Exercises 9 and 10, describe and correct the error in the first step of solving the system of linear equations.

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (318)

Question 9.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (319)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (320)

Question 10.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (321)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (322)

In Exercises 11–16, solve the system using the elimination method.

Question 11.
3x – y + 2z = 4
6x – 2y + 4z = -8
2x – y + 3z = 10
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (323)

Question 12.
5x + y – z = 6
x + y + z = 2
12x + 4y = 10
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (324)

Question 13.
x + 3y – z = 2
x + y – z = 0
3x + 2y – 3z = -1
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (325)

Question 14.
x + 2y – z = 3
-2x – y + z = -1
6x – 3y – z = -7
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (326)

Question 15.
x + 2y + 3z = 4
-3x + 2y – z = 12
-2x – 2y – 4z = -14
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (327)

Question 16.
-2x – 3y + z = -6
x + y – z = 5
7x + 8y – 6z = 31
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (328)

Question 17.
MODELING WITH MATHEMATICS Three orders are placed at a pizza shop. Two small pizzas, a liter of soda, and a salad cost $14; one small pizza, a liter of soda, and three salads cost $15; and three small pizzas, a liter of soda, and two salads cost $22. How much does each item cost?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (329)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (330)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (331)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (332)

Question 18.
MODELING WITH MATHEMATICS Sam’s Furniture Store places the following advertisem*nt in the local newspaper. Write a system of equations for the three combinations of furniture. What is the price of each piece of furniture? Explain.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (333)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (334)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (335)

In Exercises 19–28, solve the system of linear equations using the substitution method.

Question 19.
-2x + y + 6z = 1
3x + 2y + 5z = 16
7x + 3y – 4z = 11
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (336)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (337)

Question 20.
x – 6y – 2z = -8
-x + 5y + 3z = 2
3x – 2y – 4z = 18
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (338)

Question 21.
x + y + z = 4
5x + 5y + 5z = 12
x – 4y + z = 9
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (339)

Question 22.
x + 2y = -1
-x + 3y + 2z = -4
-x + y – 4z = 10
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (340)

Question 23.
2x – 3y + z = 10
y + 2z = 13
z = 5
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (341)

Question 24.
x = 4
x + y = -6
4x – 3y + 2z = 26
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (342)

Question 25.
x + y – z = 4
3x + 2y + 4z = 17
-x + 5y + z = 8
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (343)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (344)

Question 26.
2x – y – z = 15
4x + 5y + 2z = 10
-x – 4y + 3z = -20
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (345)

Question 27.
4x + y + 5z = 5
8x + 2y + 10z = 10
x – y – 2z = -2
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (346)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (347)

Question 28.
x + 2y – z = 3
2x + 4y – 2z = 6
-x – 2y + z = -6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (348)

Question 29.
PROBLEM SOLVING The number of left-handed people in the world is one-tenth the number of right-handed people. The percent of right-handed people is nine times the percent of left-handed people and ambidextrous people combined. What percent of people are ambidextrous?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (349)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (350)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (351)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (352)

Question 30.
MODELING WITH MATHEMATICS Use a system of linear equations to model the data in the following newspaper article. Solve the system to find how many athletes finished in each place.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (353)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (354)

Question 31.
WRITING Explain when it might be more convenient to use the elimination method than the substitution method to solve a linear system. Give an example to support your claim.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (355)

Question 32.
REPEATED REASONING Using what you know about solving linear systems in two and three variables, plan a strategy for how you would solve a system that has four linear equations in four variables.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (356)

MATHEMATICAL CONNECTIONS In Exercises 33 and 34, write and use a linear system to answer the question.

Question 33.
The triangle has a perimeter of 65 feet. What are the lengths of sides ℓ, m, and n?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (357)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (358)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (359)

Question 34.
What are the measures of angles A, B, and C?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (360)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (361)

Question 35.
OPEN-ENDED Consider the system of linear equations below. Choose nonzero values for a, b, and c so the system satisfies the given condition. Explain your reasoning.
x + y + z = 2
ax + by + cz = 10
x – 2y + z = 4
a. The system has no solution.
b. The system has exactly one solution.
c. The system has infinitely many solutions.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (362)

Question 36.
MAKING AN ARGUMENT A linear system in three variables has no solution. Your friend concludes that it is not possible for two of the three equations to have any points in common. Is your friend correct? Explain your reasoning.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (363)

Question 37.
PROBLEM SOLVING A contractor is hired to build an apartment complex. Each 840-square-foot unit has a bedroom, kitchen, and bathroom. The bedroom will be the same size as the kitchen. The owner orders 980 square feet of tile to completely cover the floors of two kitchens and two bathrooms. Determine how many square feet of carpet is needed for each bedroom.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (364)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (365)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (366)

Question 38.
THOUGHT PROVOKING Does the system of linear equations have more than one solution? Justify your answer.
4x + y + z = 0
2x + \(\frac{1}{2}\)y – 3z = 0
-x – \(\frac{1}{4}\)y – z = 0
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (367)

Question 39.
PROBLEM SOLVING A florist must make 5 identical bridesmaid bouquets for a wedding. The budget is $160, and each bouquet must have 12 flowers. Roses cost $2.50 each, lilies cost $4 each, and irises cost $2 each. The florist wants twice as many roses as the other two types of flowers combined.
a. Write a system of equations to represent this situation, assuming the florist plans to use the maximum budget.
b. Solve the system to find how many of each type of flower should be in each bouquet.
c. Suppose there is no limitation on the total cost of the bouquets. Does the problem still have exactly one solution? If so, find the solution. If not, give three possible solutions.
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (368)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (369)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (370)

Question 40.
HOW DO YOU SEE IT? Determine whether the system of equations that represents the circles has no solution, one solution, or infinitely many solutions. Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (371)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (372)

Question 41.
CRITICAL THINKING Find the values of a, b, and c so that the linear system shown has (-1, 2, -3) as its only solution. Explain your reasoning.
x + 2y – 3z = a
– x – y + z = b
2x + 3y – 2z = c
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (373)

Question 42.
ANALYZING RELATIONSHIPS Determine which arrangement(s) of the integers -5, 2, and 3 produce a solution of the linear system that consist of only integers. Justify your answer.
x – 3y + 6z = 21
_x + _y + _z = -30
2x – 5y + 2z = -6
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (374)

Question 43.
ABSTRACT REASONING Write a linear system to represent the first three pictures below. Use the system to determine how many tangerines are required to balance the apple in the fourth picture. Note:The first picture shows that one tangerine and one apple balance one grapefruit.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (375)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (376)

Maintaining Mathematical Proficiency

Simplify. (Skills Review Handbook)

Question 44.
(x – 2)2
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (377)

Question 45.
(3m + 1)2

Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (378)

Question 46.
(2z – 5)2
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (379)
Question 47.
(4 – y)2
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (380)
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (381)

Write a function g described by the given transformation of f(x) =∣x∣− 5.(Section 1.2)

Question 48.
translation 2 units to the left
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (382)

Question 49.
reflection in the x-axis
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (383)

Question 50.
translation 4 units up
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (384)

Question 51.
vertical stretch by a factor of 3
Answer:
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (385)

Linear Functions Performance Task: Secret of the Hanging Baskets

1.3–1.4 What Did You Learn?

Core Vocabulary
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (386)

Core Concepts
Section 1.3
Writing an Equation of a Line, p. 22
Finding a Line of Fit, p. 24
Section 1.4
Solving a Three-Variable System, p. 31
Solving Real-Life Problems, p. 33

Mathematical Practices

Question 1.
Describe how you can write the equation of the line in Exercise 7 on page 26 using only one of the labeled points.

Question 2.
How did you use the information in the newspaper article in Exercise 30 on page 35 to write a system of three linear equations?

Question 3.
Explain the strategy you used to choose the values for a, b, and c in Exercise 35 part (a) on page 35.

Performance Task

Secret of the Hanging Baskets
A carnival game uses two baskets hanging from springs at different heights. Next to the higher basket is a pile of baseballs. Next to the lower basket is a pile of golf balls. The object of the game is to add the same number of balls to each basket so that the baskets have the same height. But there is a catch—you only get one chance. What is the secret to winning the game?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (387)
To explore the answers to this question and more, go to BigIdeasMath.com.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (388)

Linear Functions Chapter Review

Graph the function and its parent function. Then describe the transformation.

Question 1.
f(x) = x + 3
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (389)

Question 2.
g(x) = | x | – 1
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (390)

Question 3.
h(x) = \(\frac{1}{2}\)x2

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (391)

Question 4.
h(x) = 4

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (392)

Question 5.
f(x) = -| x | – 3

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (393)

Question 6.
g(x) = -3(x + 3)2

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (394)

Write a function g whose graph represents the indicated transformations of the graph of f. Use a graphing calculator to check your answer.

Question 7.
f(x) = | x |; reflection in the x-axis followed by a translation 4 units to the left

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (395)

Question 8.
f(x) = | x | ; vertical shrink by a factor of \(\frac{1}{2}\) followed by a translation 2 units up

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (396)

Question 9.
f(x) = x; translation 3 units down followed by a reflection in the y-axis

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (397)

Question 10.
The table shows the total number y (in billions) of U.S. movie admissions each year for x years. Use a graphing calculator to find an equation of the line of best fit for the data.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (398)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (399)

Question 11.
You ride your bike and measure how far you travel. After 10 minutes, you travel 3.5 miles. After 30 minutes, you travel 10.5 miles. Write an equation to model your distance. How far can you ride your bike in 45 minutes?
Answer:
Given,
You ride your bike and measure how far you travel. After 10 minutes, you travel 3.5 miles. After 30 minutes, you travel 10.5 miles.
m = (10.5 – 3.5)/30 – 10
m = 7/20
y – 3.5 = 7/20(x – 10)
y – 3.5 = 7/20 x – 3.5
y = 7/20x
x = 45
y = 7/20 (45)
y = 15.75 miles

Question 12.
x + y + z = 3
-x + 3y + 2z = -8
x = 4z

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (400)

Question 13.
2x – 5y – z = 17
x + y + 3z = 19
-4x + 6y + z = -20

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (401)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (402)

Question 14.
x + y + z = 2
2x – 3y + z = 11
-3x + 2y – 2z = -13

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (403)

Question 15.
x + 4y – 2z = 3
x + 3y + 7z = 1
2x + 9y – 13z = 2

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (404)

Question 16.
x – y + 3z = 6
x – 2y = 5
2x – 2y + 5z = 9

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (405)

Question 17.
x + 2y = 4
x + y + z = 6
3x + 3y + 4z = 28

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (406)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (407)

Question 18.
A school band performs a spring concert for a crowd of 600 people. The revenue for the concert is $3150. There are 150 more adults at the concert than students. How many of each type of ticket are sold?
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (408)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (409)

Linear Functions Chapter Test

Write an equation of the line and interpret the slope and y-intercept.

Question 1.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (410)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (411)

Question 2.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (412)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (413)

Solve the system. Check your solution, if possible.

Question 3.
-2x + y + 4z = 5
x + 3y – z = 2
4x + y – 6z = 11

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (414)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (415)

Question 4.
y = \(\frac{1}{2}\)z
x + 2y + 5z = 2
3x + 6y – 3z = 9

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (416)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (417)
Question 5.
x – y + 5z = 3
2x + 3y – z = 2
-4x – y – 9z = -8
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (418)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (419)

Graph the function and its parent function. Then describe the transformation.

Question 6.

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (420)

Question 7.
f(x) = (3x)2

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (421)

Question 8.
f(x) = 4
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (422)

Match the transformation of f(x) = x with its graph. Then write a rule for g.

Question 9.
g(x) = 2f(x) + 3
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (423)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (424)

Question 10.
g(x) = 3f(x) – 2
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (425)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (426)

Question 11.
g(x) = -2f(x) – 3
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (427)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (428)

Question 12.
A bakery sells doughnuts, muffins, and bagels. The bakery makes three times as many doughnuts as bagels. The bakery earns a total of $150 when all 130 baked items in stock are sold. How many of each item are in stock? Justify your answer.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (429)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (430)

Question 13.
A fountain with a depth of 5 feet is drained and then refilled. The water level (in feet) after t minutes can be modeled by f(t) = \(\frac{1}{4}\)|t – 20 |. A second fountain with the same depth is drained and filled twice as quickly as the first fountain. Describe how to transform the graph of f to model the water level in the second fountain after t minutes. Find the depth of each fountain after 4 minutes. Justify your answers.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (431)

Linear Functions Cumulative Assessment

Question 1.
Describe the transformation of the graph of f(x) = 2x – 4 represented in each graph.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (432)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (433)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (434)
Question 2.
The table shows the tuition costs for a private school between the years 2010 and 2013.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (435)
a. Verify that the data show a linear relationship. Then write an equation of a line of fit.
b. Interpret the slope and y-intercept in this situation.
c. Predict the cost of tuition in 2015.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (436)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (437)

Question 3.
Your friend claims the line of best fit for the data shown in the scatter plot has a correlation coefficient close to 1. Is your friend correct? Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (438)
Answer:
From the graph, the correlation coefficient is r = -0.86
So, my friend is not correct, since the correlation coefficient is close to -1.

Question 4.
Order the following linear systems from least to greatest according to the number of solutions.
A. 2x + 4y – z = 7
14x + 28y – 7z = 49
-x + 6y + 12z = 13
B. 3x – 3y + 3z = 5
-x + y – z = 5
-x + y – z = 8
14x – 3y + 12z = 108
C. 4x – y + 2z = 18
-x + 2y + z = 11
3x + 3y – 4z = 44
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (439)
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (440)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (441)

Question 5.
You make a DVD of three types of shows: comedy, drama, and reality-based. An episode of a comedy lasts 30 minutes, while a drama and a reality-based episode each last 60 minutes. The DVDs can hold 360 minutes of programming.
a. You completely fill a DVD with seven episodes and include twice as many episodes of a drama as a comedy. Create a system of equations that models the situation.
b. How many episodes of each type of show are on the DVD in part (a)?
c. You completely fill a second DVD with only six episodes. Do the two DVDs have a different number of comedies? dramas? reality-based episodes? Explain.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (442)

Question 6.
The graph shows the height of a hang glider over time. Which equation models the situation?
A. y + 450 = 10x
B. 10y = -x+ 450
C. \(\frac{1}{10}\)y = -x + 450
D. 10x + y = 450
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (443)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (444)

Question 7.
Let f(x) = x and g(x) = -3x – 4. Select the possible transformations (in order) of the graph of f represented by the function g.
A. reflection in the x-axis
B. reflection in the y-axis
C. vertical translation 4 units down
D. horizontal translation 4 units right
E. horizontal shrink by a factor of \(\frac{1}{3}\)
F. vertical stretch by a factor of 3
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (445)

Question 8.
Choose the correct equality or inequality symbol which completes the statement below about the linear functions f and g. Explain your reasoning.
Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (446)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (447)

Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions (2024)

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