# Big Ideas Math Algebra 2 Answers Chapter 1 Linear Functions

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

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### 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 + $$\frac{3}{5}$$
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)
$$\frac{1}{2}$$(22+ 22)
$$\frac{1}{2}$$(4+ 22)
$$\frac{1}{2}$$(26) = 13

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

$$\frac{1}{6}$$(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.

Question 8.
Reflect the triangle in the y-axis. Then translate 2 units left.

Question 9.
Translate the trapezoid 3 units down. Then reflect in the x-axis.

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.

The order of operations tells us the order to solve steps in expressions with more than one operation. First, we solve any operations 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

Question 2.
y = | x + 2 |

Question 3.
y = -x2 + 1

Question 4.
y = $$\sqrt{x-1}$$

Question 5.
y = x3 – 2

Question 6.
y = 0.25x3

Determine whether the viewing window is square. Explain.

Question 7.
-8 ≤ x ≤ 8, -2 ≤ y ≤ 8
-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
-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
-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
-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
-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
-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 a function is 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.

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 and range are 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.

a. The equation for the given graph of absolute value function in exploration 1 is y = |x|

b. y = √x

c. y = c

y = e^x

y = x³

y = x

y = 1/x

y = x²

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.

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

Question 2.
g(x) = x + 3

Question 3.
h(x) = (x – 2)2

Question 4.
n(x) = – | x |

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

Question 5.
g(x) = 3x

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

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

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

Question 9.
d(x) = 3(x – 5)2 – 1

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?

### 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.

Question 2.
DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

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.

Question 4.

Question 5.

Question 6.

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.

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: In Exercises 9–18, graph the function and its parent function. Then describe the transformation. Question 9. g(x) = x + 4 Answer: Question 10. f(x) = x – 6 Answer: Question 11. f(x) = x2 – 1 Answer: Question 12. h(x) = (x+ 4)2 Answer: Question 13. g(x) = | x – 5 | Answer: Question 14. f(x) = 4 + | x | Answer: Question 15. h(x) = -x2 Question 16. g(x) = -x Answer: Question 17. f(x) = 3 Answer: Question 18. f(x) = -2 Answer: In Exercises 19–26, graph the function and its parent function. Then describe the transformation. Question 19. f(x) = $$\frac{1}{3}$$x Answer: Question 20. g(x) = 4x Answer: Question 21. f(x) = 2x2 Answer: Question 22. h(x) = $$\frac{1}{3}$$x2 Answer: Question 23. h(x) = $$\frac{3}{4}$$x Answer: Question 24. g(x) = $$\frac{4}{3}$$x Answer: Question 25. h(x) = 3 | x | Answer: Question 26. f(x) = $$\frac{1}{2}$$ | x | Answer: 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: Question 28. h(x) = -x + 5 Answer: Question 29. h(x) = -3 | x | – 1 Answer: Question 30. f(x) = $$\frac{3}{4}$$ | x | + 1 Answer: Question 31. g(x) = $$\frac{1}{2}$$x2 – 6 Answer: Question 32. f(x) = 4x2 – 3 Answer: Question 33. f(x) = -(x + 3)2 + $$\frac{1}{4}$$ Answer: Question 34. g(x) = – | x – 1 | – $$\frac{1}{2}$$ Answer: ERROR ANALYSIS In Exercises 35 and 36, identify and correct the error in describing the transformation of the parent function. Question 35. Answer: Question 36. Answer: MATHEMATICAL CONNECTIONS In Exercises 37 and 38, find the coordinates of the figure after the transformation. Question 37. Translate 2 units down. Answer: Question 38. Reflect in the x-axis. Answer: 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: Question 40. h(x) = | x – 3 | + 2 Answer: Question 41. g(x) = 3x + 4 Answer: Question 42. f(x) = -4x + 11 Answer: Question 43. f(x) = 5x2 – 2 Answer: Question 44. f(x) = -2x2 + 6 Answer: 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. Answer: 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: Question 47. USING STRUCTURE Graph the functions f(x) = | x – 4 | and g(x) = | x | – 4. Are they equivalent? Explain. Answer: Question 48. HOW DO YOU SEE IT? Consider the graphs of f, g, and h. 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: 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: 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. Answer: 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: 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. Answer: 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: 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. Answer: 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: Question 56. f(x) = | x | – 3; (-2, -5) Answer: Question 57. f(x) = x – 3; (5, 2) Answer: Question 58. f(x) = x – 4; (12, 8) Answer: Find the x-intercept and the y-intercept of the graph of the equation. (Skills Review Handbook) Question 59. y = x Answer: 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: Question 62. x – 2y = 8 Answer: ### 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 y = | x | + k Transformation 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 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 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 = $$[\sqrt{x}/latex] – 4 b. y = [latex][\sqrt{x + 4}/latex] c. y = –[latex][\sqrt{x}/latex] d. y = x2 + 1 e. y = (x – 1)2 f. y = -x2 1.2 Lesson Monitoring Progress Write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer. Question 1. f(x) = 3x; translation 5 units up Question 2. f(x) = | x | – 3; translation 4 units to the right Question 3. f(x) = – | x + 2 | – 1; reflection in the x-axis Question 4. f(x) = [latex]\frac{1}{2}$$x+ 1; reflection in the y-axis 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 Question 6. f(x) = | x | – 3; vertical shrink by a factor of $$\frac{1}{3}$$ 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. Question 8. WHAT IF? In Example 5, your revenue function is f(x) = 3x. How does this affect your profit for 100 downloads? ### 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: Question 2. WHICH ONE DOESN’T BELONG? Which transformation does not belong with the other three? Explain your reasoning. Answer: 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: Question 4. f(x) = x + 2; translation 2 units to the right Answer: Question 5. f(x) = | 4x + 3 | + 2; translation 2 units down Answer: Question 6. f(x) = 2x – 9; translation 6 units up Answer: Question 7. f(x) = 4 – | x + 1 | Answer: Question 8. f(x) = | 4x | + 5 Answer: Question 9. WRITING Describe two different translations of the graph of f that result in the graph of g. Answer: 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?

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

Question 12.
f(x) = $$\frac{1}{2}$$x – 3; reflection in the x-axis

Question 13.
f(x) = | 6x | – 2; reflection in the y-axis

Question 14.
f(x) = | 2x – 1 | + 3; reflection in the y-axis

Question 15.
f(x) = -3 + | x – 11 |; reflection in the y-axis

Question 16.
f(x) = -x+ 1; reflection in the y-axis

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

Question 18.
f(x) = 2x+ 6; vertical shrink by a factor of $$\frac{1}{2}$$

Question 19.
f(x) = | 2x | + 4; horizontal shrink by a factor of $$\frac{1}{2}$$

Question 20.
f(x) = | x+ 3 | ; horizontal stretch by a factor of 4

Question 21.
f(x) = -2 | x – 4 | + 2

Question 22.
f(x) = 6 – x

ANALYZING RELATIONSHIPS In Exercises 23–26, match the graph of the transformation of f with the correct equation shown. Explain your reasoning.

Question 23.

Question 24.

Question 25.

Question 26.

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

Question 28.
f(x) = x; translation 3 units down followed by a vertical shrink by a factor of $$\frac{1}{3}$$

Question 29.
f(x) = | x | ; translation 2 units to the right followed by a horizontal stretch by a factor of 2

Question 30.
f(x) = | x |; reflection in the y-axis followed by a translation 3 units to the right

Question 31.
f(x) = | x |

Question 32.
f(x) = | x |

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.

Question 34.

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.

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.

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 |

Question 38.
f(x) = – | x | – 2

Question 39.
f(x) = -x + 4

Question 40.
f(x) = x – 5

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.

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.

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.

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.

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.

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.

Maintaining Mathematical Proficiency

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

Question 46.
f(x) = x + 4; x = 3

Question 47.
f(x) = 4x – 1; x = -1

Question 48.
f(x) = -x + 3; x = 5

Question 49.
f(x) = -2x – 2; x = -1

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

Question 50.

Question 51.

### Linear Functions Study Skills Taking Control of Your Class Time

1.1 – 1.2 What Did You Learn?

Core Vocabulary

Core Concepts

Section 1.1

Section 1.2

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. 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. Question 2. Question 3. Graph the function and its parent function. Then describe the transformation. (Section 1.1) Question 4. f(x) = $$\frac{3}{2}$$ Question 5. f(x) = 3x Answer: Question 6. f(x) = 2(x – 1)2 Answer: Question 7. f(x) = – | x + 2 | – 7 Answer: Question 8. f(x) = $$\frac{1}{4}$$x2 + 1 Answer: Question 9. f(x) = –$$\frac{1}{2}$$x – 4 Answer: 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) 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)
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.

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

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.

Question 3.
How can you use a linear function to model and analyze a real-life situation?
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.
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.

 Speed Time (in second) 100 0 90 1 80 2 70 3 60 4 50 5 40 6 30 7

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

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?

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.

Question 3.
The table shows the humerus lengths (in centimeters) and heights (in centimeters) of several females.

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).

### 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.

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?

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.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.
MODELING WITH MATHEMATICS Two newspapers charge a fee for placing an advertisement in their paper plus a fee based on the number of lines in the advertisement. The table shows the total costs for different length advertisements at the Daily Times. The total cost y (in dollars) for an advertisement 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 advertisement for the total costs to be the same?

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?

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

Question 11.

Question 12.

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.

Question 14.

Question 15.

Question 16.

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.

Question 18.

### 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)__________.

Question 2.
WRITING Explain how you know when a linear system in three variables has infinitely many solutions.

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

Question 4.
x + 4y – 6z = -1
2x – y + 2z = -7
-x + 2y – 4z = 5
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

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

Question 7.
2x + 2y + 5z = -1
2x – y + z = 2
2x + 4y – 3z = 14

Question 8.
3x + 2y – 3z = -2
7x – 2y + 5z = -14
2x + 4y + z = 6

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

Question 9.

Question 10.

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

Question 12.
5x + y – z = 6
x + y + z = 2
12x + 4y = 10

Question 13.
x + 3y – z = 2
x + y – z = 0
3x + 2y – 3z = -1

Question 14.
x + 2y – z = 3
-2x – y + z = -1
6x – 3y – z = -7

Question 15.
x + 2y + 3z = 4
-3x + 2y – z = 12
-2x – 2y – 4z = -14

Question 16.
-2x – 3y + z = -6
x + y – z = 5
7x + 8y – 6z = 31

Question 17.

### Linear Functions Chapter Test

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

Question 1.

Question 2.

Solve the system. Check your solution, if possible.

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

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

Question 5.
x – y + 5z = 3
2x + 3y – z = 2
-4x – y – 9z = -8

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

Question 6.

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

Question 8.
f(x) = 4

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

Question 9.
g(x) = 2f(x) + 3

Question 10.
g(x) = 3f(x) – 2

Question 11.
g(x) = -2f(x) – 3

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.

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.

### Linear Functions Cumulative Assessment

Question 1.
Describe the transformation of the graph of f(x) = 2x – 4 represented in each graph.

Question 2.
The table shows the tuition costs for a private school between the years 2010 and 2013.

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.

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.

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

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.

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

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

Question 8.
Choose the correct equality or inequality symbol which completes the statement below about the linear functions f and g. Explain your reasoning.

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