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

Scoring good grades in any of your exams can make you achieve what you are looking for but improving your subject knowledge make you invent a new thing that you dreamt of. So, learn and practice all the concepts covered in * BIM Algebra 2 Answer Key Ch 1 Linear Functions* to enhance subject knowledge and increase your math skills. Big Ideas Math Book Algebra 2 Chapter 1 Solutions is the best material for you all to prepare & score well in the exams. Therefore, click on the links and kickstart your preparation.

- 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 • 2^{3} + 7

Question 2.

4 – 2(3 + 2)^{2}

Question 3.

48 ÷ 4^{2} + \(\frac{3}{5}\)

Question 4.

50 ÷ 5^{2} • 2

Question 5.

\(\frac{1}{2}\)(2^{2}+ 22)

Question 6.

\(\frac{1}{6}\)(6 + 18) – 2^{2}

**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.

### 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 = -x^{2} + 1

Question 4.

y = \(\sqrt{x-1}\)

Question 5.

y = x^{3} – 2

Question 6.

y = 0.25x^{3}

**Determine whether the viewing window is square. Explain.**

Question 7.

-8 ≤ x ≤ 8, -2 ≤ y ≤ 8

Question 8.

-7 ≤ x ≤ 8, -2 ≤ y ≤ 8

Question 9.

-6 ≤ x ≤ 9, -2 ≤ y ≤ 8

Question 10.

-2 ≤ x≤ 2, -3 ≤ y ≤ 3

Question 11.

-4 ≤ x ≤ 5, -3 ≤ y ≤ 3

Question 12.

-4 ≤ x ≤ 4, -3 ≤ y ≤ 3

### Lesson 1.1 Parent Functions and Transformations

**Essential Question**

What are the characteristics of some of the basic parent functions?

**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.

**Communicate Your Answer**

Question 2.

What are the characteristics of some of the basic parent functions?

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.

**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) = \(\frac{3}{2}\)x^{2}

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) = x^{2} is the ______ of f(x) = 2x^{2} – 3.

Answer:

Question 2.

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

Answer:

**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.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

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:

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 – 250x^{2}. 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) = x^{2} – 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) = -x^{2}

Answer:

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) = 2x^{2
}

Answer:

Question 22.

h(x) = \(\frac{1}{3}\)x^{2
}

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) = \(\frac{1}{2}\)x^{2} – 6

Answer:

Question 32.

f(x) = 4x^{2} – 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) = 5x^{2} – 2

Answer:

Question 44.

f(x) = -2x^{2} + 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) = -16t^{2} + 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) = 4x^{2
}

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:

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?

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 = x^{2} + 1

e. y = (x – 1)^{2}

f. y = -x^{2}

**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?

Answer:

**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:

Question 12.

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

Answer:

Question 13.

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

Answer:

Question 14.

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

Answer:

Question 15.

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

Answer:

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

Answer:

Question 18.

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

Answer:

Question 19.

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

Answer:

Question 20.

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

Answer:

Question 21.

f(x) = -2 | x – 4 | + 2

Answer:

Question 22.

f(x) = 6 – x

Answer:

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

Question 23.

Answer:

Question 24.

Answer:

Question 25.

Answer:

Question 26.

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:

Question 28.

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

Answer:

Question 29.

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

Answer:

Question 30.

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

Answer:

Question 31.

f(x) = | x |

Answer:

Question 32.

f(x) = | x |

Answer:

**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.

Answer:

Question 34.

Answer:

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:

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:

**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 |

Answer:

Question 38.

f(x) = – | x | – 2

Answer:

Question 39.

f(x) = -x + 4

Answer:

Question 40.

f(x) = x – 5

Answer:

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:

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.

Answer:

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:

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:

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:

**Maintaining Mathematical Proficiency**

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

Question 46.

f(x) = x + 4; x = 3

Answer:

Question 47.

f(x) = 4x – 1; x = -1

Answer:

Question 48.

f(x) = -x + 3; x = 5

Answer:

Question 49.

f(x) = -2x – 2; x = -1

Answer:

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

Question 50.

Answer:

Question 51.

Answer:

### 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

Question 6.

f(x) = 2(x – 1)^{2}

Question 7.

f(x) = – | x + 2 | – 7

Question 8.

f(x) = \(\frac{1}{4}\)x^{2} + 1

Question 9.

f(x) = –\(\frac{1}{2}\)x – 4

**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

Question 11.

f(x) = -3 | x – 4 | ; vertical shrink by a factor of \(\frac{1}{2}\)

Question 12.

f(x) = 3 | x + 5 |; reflection in the x-axis

Question 13.

f(x) = \(\frac{1}{3}\)x – \(\frac{2}{3}\) ; translation 4 units left

**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.

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.

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 |.

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 |.

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)

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)

### 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.

**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.

**Communicate Your Answer**

Question 3.

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

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.

**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.

Answer:

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:

**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.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.

Answer:

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?

Answer:

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:

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

Question 11.

Answer:

Question 12.

Answer:

**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.

Answer:

Question 14.

Answer:

Question 15.

Answer:

Question 16.

Answer:

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.

Answer:

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.

Answer:

**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.

Answer:

Question 20.

Answer:

Question 21.

Answer:

Question 22.

Answer:

Question 23.

Answer:

Question 24.

Answer:

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:

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.

a. What is the slope of the line? What does the slope represent?

b. What is the domain and range of the function? What does each represent?

c. How much do you still owe after making payments for 12 months?

Answer:

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:

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:

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:

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:

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?

Answer:

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.

Answer:

**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:

Question 34.

4x + 3y = 2

2x – 3y = 1

Answer:

Question 35.

2x + 2y = 3

x = 4y – 1

Answer:

Question 36.

y = 1 + x

2x + y = -2

Answer:

Question 37.

\(\frac{1}{2}\)x + 4y = 4

2x – y = 1

Answer:

Question 38.

y = x – 4

4x + y = 26

Answer:

### 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

b. 2x – 3y = 3

x + 2y = 5

c. 2x – 3y = 3

-4x + 6y = 6

**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

b. x+ 3y = 1

-x + 2y = 4

c. x + y = 0

3x + 2y = 1

**Communicate Your Answer**

Question 3.

How can you determine the number of solutions of a linear system?

Question 4.

Suppose you were given a system of three linear equations in three variables. Explain how you would approach solving such a system.

Question 5.

Apply your strategy in Question 4 to solve the linear system.

**1.4 Lesson**

**Monitoring Progress**

Question 1.

x – 2y + z = -11

3x + 2y – z = 7

-x + 2y + 4z = -9

Question 2.

x + y – z = -1

4x + 4y – 4z = -2

3x + 2y + z = 0

Question 3.

x + y + z = 8

x – y + z = 8

2x + y + 2z = 16

Question 4.

In Example 3, describe the solutions of the system using an ordered triple in terms of y.

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?

### 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:

Question 2.

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

Answer:

**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:

Question 4.

x + 4y – 6z = -1

2x – y + 2z = -7

-x + 2y – 4z = 5

Answer:

Question 5.

2x + y – z = 9

-x + 6y + 2z = -17

5x + 7y + z = 4

Answer:

Question 6.

3x + 2y – z = 8

-3x + 4y + 5z = -14

x – 3y + 4z = -14

Answer:

Question 7.

2x + 2y + 5z = -1

2x – y + z = 2

2x + 4y – 3z = 14

Answer:

Question 8.

3x + 2y – 3z = -2

7x – 2y + 5z = -14

2x + 4y + z = 6

Answer:

**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.

Answer:

Question 10.

Answer:

**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:

Question 12.

5x + y – z = 6

x + y + z = 2

12x + 4y = 10

Answer:

Question 13.

x + 3y – z = 2

x + y – z = 0

3x + 2y – 3z = -1

Answer:

Question 14.

x + 2y – z = 3

-2x – y + z = -1

6x – 3y – z = -7

Answer:

Question 15.

x + 2y + 3z = 4

-3x + 2y – z = 12

-2x – 2y – 4z = -14

Answer:

Question 16.

-2x – 3y + z = -6

x + y – z = 5

7x + 8y – 6z = 31

Answer:

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?

Answer:

Question 18.

**MODELING WITH MATHEMATICS** Sam’s Furniture Store places the following advertisement 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.

Answer:

**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:

Question 20.

x – 6y – 2z = -8

-x + 5y + 3z = 2

3x – 2y – 4z = 18

Answer:

Question 21.

x + y + z = 4

5x + 5y + 5z = 12

x – 4y + z = 9

Answer:

Question 22.

x + 2y = -1

-x + 3y + 2z = -4

-x + y – 4z = 10

Answer:

Question 23.

2x – 3y + z = 10

y + 2z = 13

z = 5

Answer:

Question 24.

x = 4

x + y = -6

4x – 3y + 2z = 26

Answer:

Question 25.

x + y – z = 4

3x + 2y + 4z = 17

-x + 5y + z = 8

Answer:

Question 26.

2x – y – z = 15

4x + 5y + 2z = 10

-x – 4y + 3z = -20

Answer:

Question 27.

4x + y + 5z = 5

8x + 2y + 10z = 10

x – y – 2z = -2

Answer:

Question 28.

x + 2y – z = 3

2x + 4y – 2z = 6

-x – 2y + z = -6

Answer:

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?

Answer:

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.

Answer:

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:

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:

**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?

Answer:

Question 34.

What are the measures of angles A, B, and C?

Answer:

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:

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:

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.

Answer:

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:

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:

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.

Answer:

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:

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:

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.

Answer:

**Maintaining Mathematical Proficiency**

**Simplify. **(Skills Review Handbook)

Question 44.

(x – 2)^{2
}

Answer:

Question 45.

(3m + 1)^{2
}

Answer:

Question 46.

(2z – 5)^{2
}

Answer:

Question 47.

(4 – y)^{2
}

Answer:

**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:

Question 49.

reflection in the x-axis

Answer:

Question 50.

translation 4 units up

Answer:

Question 51.

vertical stretch by a factor of 3

Answer:

### Linear Functions Performance Task: Secret of the Hanging Baskets

**1.3–1.4 What Did You Learn?**

**Core Vocabulary**

**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?

To explore the answers to this question and more, go to BigIdeasMath.com.

### Linear Functions Chapter Review

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

Question 1.

f(x) = x + 3

Question 2.

g(x) = | x | – 1

Question 3.

h(x) = \(\frac{1}{2}\)x^{2}

Question 4.

h(x) = 4

Question 5.

f(x) = -| x | – 3

Question 6.

g(x) = -3(x + 3)^{2}

**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

Question 8.

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

Question 9.

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

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.

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?

Question 12.

x + y + z = 3

-x + 3y + 2z = -8

x = 4z

Question 13.

2x – 5y – z = 17

x + y + 3z = 19

-4x + 6y + z = -20

Question 14.

x + y + z = 2

2x – 3y + z = 11

-3x + 2y – 2z = -13

Question 15.

x + 4y – 2z = 3

x + 3y + 7z = 1

2x + 9y – 13z = 2

Question 16.

x – y + 3z = 6

x – 2y = 5

2x – 2y + 5z = 9

Question 17.

x + 2y = 4

x + y + z = 6

3x + 3y + 4z = 28

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?

### 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.

f(x) = | x – 1 |

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.

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.