5-1 rate of change and slope - somersetcanyons.com 5... · compare and contrast how does finding a...

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Vocabulary

Review

Chapter 5 286

5-1 Rate of Change and Slope

1. Circle the rate that matches this situation: Ron reads 5 books every 2 weeks.

5 weeks2 books

2 books5 weeks

5 books2 weeks

2. Write always, sometimes, or never.

A rate is 9 a ratio.

A ratio is 9 a rate.

3. Underline the correct word to complete each sentence.

A rate compares two quantities by division / multiplication .

A rate compares quantities in different / the same unit(s).

Vocabulary Builder

slope (noun) slohp

Definition: Slope is the ratio of the vertical change (or rise) to the horizontal change (or run) between two points on a line. Slope is also called the rate of change.

Main Idea: Slope describes the steepness of a line in the coordinate plane.

Examples: You can measure the slope of a hill, mountain, road, or roof.

Use Your Vocabulary

4. How does the slope of a road affect a person’s driving?

_______________________________________________________________________

_______________________________________________________________________

5. What kind of ski slope would a beginner skier use?

_______________________________________________________________________

_______________________________________________________________________

vertical changehorizontal change

riserunslope

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d.Problem 1

Problem 2

x

y

O2

2

4

4

2 4

4

1

2

3

4

260

520

780

1040

Time(min)

Distance(ft)

Distance Marched

287 Lesson 5-1

Finding Rate of Change Using a Table

Got It? The table at the right shows the distance a band marches over time. The rate of change from one row of the table to the next is 260 feet per minute. Do you get the rate of change of 260 feet per minute if you use nonconsecutive rows of the table? Explain.

6. Use the values from the second and fourth rows to find the rate of change.

rate of change 5change in distance

change in time

52 520

4 2

52

51

When you use nonconsecutive rows, the rate of change is ft per min.

7. Is the rate of change you found in Exercise 6 the same as if you had used two consecutive rows? Explain why or why not.

_______________________________________________________________________

_______________________________________________________________________

Finding Slope Using a Graph

Got It? What is the slope of the line?

8. Label each point on the graph with its coordinates.

9. Draw a vertical arrow to represent the rise.

rise 5

10. Draw a horizontal arrow to represent the run.

run 5

11. Underline the correct word(s) to complete the sentence.

Because the points are on the same line,the rate of change from point to point

is constant / differs .

12. Write the slope of the line.

slope 5vertical change

horizontal change5

rise

run5

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Problem 4

Problem 3

x

y

O 42

2

4

24

2

4

Key Concept The Slope Formula

x2 x1

y2 y1B(x2, y2)

A(x1, y1)

Chapter 5 288

Finding Slope Using Points

Got It? What is the slope of the line through (1, 3) and (4, 21)?

15. You can use either pair for (x2, y2).

For example, use (4, ) for (x2, y2). Then use (1, ) for (x1, y1).

16. Complete the equation.

slope 5y2 2 y1x2 2 x1

521 2

4 25

17. The slope of the line through (1, 3) and (4, 21) is .

Finding Slopes of Horizontal and Vertical Lines

Got It? What is the slope of the line through (4, 23) and (4, 2)?

18. Graph the points (4, 23) and (4, 2) and draw the line that goes through the points.

19. Is the line that you drew horizontal or vertical?

____________________________________________________

20. What is the slope of the line through (4, 23) and (4, 2)?

____________________________________________________

In the diagram, (x1, y1) are the coordinates of point A, and (x2, y2) are the coordinates of point B. To fi nd the slope of

*AB), you can use the slope formula.

slope 5 riserun 5

y2 2 y1x2 2 x1

, where x2 2 x1 2 0

When using the slope formula, the x–coordinate you use fi rst in the denominator must belong to the same ordered pair as the y–coordinate you use fi rst in the numerator.

13. To find the change in x– or y–coordinates, do you add or subtract?

_______________________________________________________________________

14. What number will you get in the denominator if the x-coordinates are the same? Explain how that will affect the answer you find for the slope.

_______________________________________________________________________

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Lesson Check

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

Check off the vocabulary words that you understand.

rate of change slope

Rate how well you can fi nd the slope of a line.

y

xO

y

xO

y

xO

y

xO

Concept Summary Slopes of Lines

289 Lesson 5-1

• Do you UNDERSTAND?

Compare and Contrast How does finding a line’s slope by counting units of vertical and horizontal change on a graph compare with finding it using the slope formula?

22. Describe how the two methods of finding slope are the same.

______________________________________________________________________________________

______________________________________________________________________________________

23. Describe how the two methods of finding slope are different.

______________________________________________________________________________________

______________________________________________________________________________________

21. Label each graph with one of the descriptions below.

negative slope positive slope slope of 0 undefined slope

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Name Class Date

5-1 Think About a PlanRate of Change and Slope

Profi t John’s business made $4500 in January and $8600 in March. What is the rate of change in his profi t for this time period?

Understanding the Problem

1. What is the formula for fi nding rate of change?

2. What are the two changing quantities that aff ect rate of change in this problem? What are the units of each quantity?

3. Will the rate of change be positive or negative? Explain.

Planning the Solution

4. Which quantity is the dependent variable? Which quantity is the independent variable? Explain.

5. What is the general equation that represents the rate of change?

Getting an Answer

6. Substitute values into your general equation and simplify. Show your work.

7. If you were to graph this relationship, what would the rate of change be in relation to your graph?

Chapter 5 290

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d.Name Class Date

5-1 Practice Form K

Rate of Change and Slope

Each rate of change is constant. Find the rate of change and explain what it represents.

1. 2.

Find the slope of each line.

3. 4.

5. 6.

Find the slope of the line that passes through each pair of points.

7. (24, 5), (1, 1) 8. (0, 0), (21, 3)

9. (2, 2), (3, 4) 10. (5, 3), (22, 24)

Find the slope of each line.

11. 12.

Fences Painted

Hours Fences

63

912

1234

x

y

O2

2

2

2

x

y

O2

2

2

2

x

y

O2

2

2

2

Miles Per Hour

Hours Miles

42

68

70140210280

x

y

O2

2

2

2

x

y

O2

2

2

2

x

y

O2

2

2

2

291 Lesson 5-1

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Name Class Date

5-1 Practice (continued) Form K

Rate of Change and Slope

Without graphing, tell whether the slope of a line that models each linear relationship is positive, negative, zero, or undefi ned. Th en fi nd the slope.

13. Th e cost of a pair of jeans is $22.50 for 1 pair and $67.50 for 3 pairs.

14. An employee earns $28.50 after 3 hours and $237.50 after 25 hours.

State the independent variable and the dependent variable in each situation. Th en fi nd the rate of change for each situation.

15. Th e cost of three gallons of milk is $8.85 and fi ve gallons of milk is $14.75.

16. Jacques fi lled 10 envelopes in 1 minute and 100 envelopes in 10 minutes.

Find the slope of the line that passes through each pair of points.

17. (7, 21), (7, 1) 18. (3, 22), (22.5, 9)

19. Q13 ,

25R, Q2

13 , 35R 20. Q2

34 , 23R, Q2

34 , 53R

21. Writing Explain why the slope of a vertical line is always undefi ned.

22. Writing Describe how to draw a line that passes through the origin and has a

slope of 35.

Each pair of points lies on a line with the given slope. Find x or y.

23. (2, 2), (5, y); slope 5 2 24. (9, 4), (x, 6); slope 5 2 13

Chapter 5 292

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5-1 Standardized Test PrepRate of Change and Slope

Multiple Choice

For Exercises 1–5, choose the correct letter.

1. What is the slope of the line that passes through the points (22, 5) and (1, 4)?

A. 23 B. 21 C. 2 13 D. 1

3

2. A line has slope 2 53. Th rough which two points could this line pass?

F. (12, 13), (17, 10) H. (0, 7), (3, 10) G. (16, 15), (13, 10) I. (11, 13), (8, 18)

3. Th e pair of points (6, y) and (10, 21) lie on a line with slope 14. What is the value of y?

A. 25 B. 22 C. 2 D. 5

4. What is the slope of a vertical line? F. 21 G. 0 H. 1 I. undefi ned

5. Shawn needs to read a book that is 374 pages long. Th e graph shown at the right shows his progress over the fi rst 5 hours of reading. If he continues to read at the same rate, how many hours total will it take for Shawn to read the entire book?

A. 15 hours C. 19 hours B. 17 hours D. 21 hours

Short Response

6. Robi has run the fi rst 4 miles of a race in 30 minutes. She reached the 6 mile point after 45 minutes. Without graphing, is the slope of the line that represents this situation positive, negative, zero, or undefi ned? What is the slope?

xO

y175

125

75100

150

2550 (2, 44)

(8, 176)

2 4 6 8 101214Hours Reading

Page

s Re

ad

293 Lesson 5-1

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Vocabulary

Review

Chapter 5 294

5-2 Direct Variation

1. Cross out the expression below that does NOT show a formula for slope.

horizontal changevertical change

y2 2 y1x2 2 x1

riserun

2. Underline the correct word in each sentence about slope.

The slope of a horizontal line is undefined / zero .

The slope of a vertical line is undefined / zero .

Vocabulary Builder

direct (adjective) duh REKT

Definition: Direct means straightforward in language or action.

Other Word Forms: directly (adverb), direction(s) (noun)

Math Usage: If the ratio of two variables is constant, then the variables form a direct variation.

What It Means: In a direct variation, one variable directly aff ects another by multiplying it by a constant value.

Both variables decrease: Th e more expensive the car, the more sales tax you pay.

One variable increases, the other variable decreases: As a candle burns longer, its height gets smaller.

Use Your Vocabulary

Choose the correct word from the list to complete each sentence.

directly direct directions

3. Renee gave the visitor 9 to the museum.

4. The fans went 9 to their seats.

5. There is a 9 connection between the outside temperature and the number of people at the beach.

y kx, where k 0, is adirect variation.

In the above, k is calledthe constant of variation.

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Problem 1

Problem 2

Think Write

I start with the function form of direct variation. y x

Next, I write an equation by substituting for k. y x

Finally, I determine the value of y when x 15. y

Now I divide each side by to solve for k.

( 2)Then I substitute 10 for y and 2 for . 10

295 Lesson 5-2

Identifying a Direct Variation

Got It? Does 4x 1 5y 5 0 represent a direct variation? If so, find the constant of variation.

6. Circle the equation that shows direct variation.

y 5 kx y 5 kx yx 5 k

7. Complete the steps to solve 4x 1 5y 5 0 for y.

4x 1 5y 5 0 Write the original equation.

5y 5 0 2 Subtract from each side.

y 5 Divide each side by .

8. Does 4x 1 5y 5 0 represent a direct variation? Explain.

_______________________________________________________________________

_______________________________________________________________________

9. In the equation 4x 1 5y 5 0, is the constant of variation.

Writing a Direct Variation Equation

Got It? Suppose y varies directly with x, and y 5 10 when x 522 . What direct variation equation relates x and y? What is the value of y when x 5 215?

10. Complete the reasoning model below.

A function in the form y 5 kx, where k 2 0, represents a direct variation. Th e constant of variation k is the coeffi cient of x.

To determine whether an equation represents a direct variation, solve it for y. If you can write the equation in the form y 5 kx, where k 2 0, it represents a direct variation.

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Problem 3

x

y 0 0

y 25

0

25

50

100

125

y 50

y 100

y 125

y x

O 20015010050

2

6

10

14

18

22

26

30

x

y

Concept Summary Graphs of Direct Variations

x

y

k 0

x

y

k 0

Chapter 5 296

Graphing a Direct Variation

Got It? Weight on the moon y varies directly with weight on Earth x. A person who weighs 100 lb on Earth weighs 16.6 lb on the moon. What is an equation that relates weight on Earth x and weight on the moon y? What is the graph of this equation?

11. Find the value of k. Round k to the nearest hundredth if necessary.

y 5 kx

5 k ?

5 k

12. To the nearest hundredth, k 5 . So, y < ? x.

13. Make a table of values. 14. Graph the values from the table.

Th e graph of a direct variation equation y 5 kx is a line with the following properties.

• The line passes through (0, 0).

• The slope of the line is k.

15. Substitute x 5 0 and y 5 0 in the equation 22x 1 y 5 3.

22x 1 y 5 3

22 ? 1 5 3

1 5 3

0 3

16. Because the graph of 22x 1 y 5 3 passes / does not pass through (0, 0), the

equation is / is not a direct variation.

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d.Problem 4

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

Lesson Check

297 Lesson 5-2

Vocabulary Determine whether each statement is always, sometimes, or never true.

Th e ordered pair (0, 0) is a solution of the direct variation equation y 5 kx.

20. Substitute (0, 0) into y 5 kx. 21. The statement is 9 true.

0 k ?

You can write a direct variation in the form y 5 k 1 x, where k u 0.

22. Is y 5 k 1 x of the form y 5 kx? 23. The statement is 9 true.

Yes / No

Th e constant of variation for a direct variation represented by y 5 kx is yx .

24. When you divide each side of y 5 kx 25. Because you cannot divide by 0,

by x, you obtain k 5 . the statement is 9 true.

Writing a Direct Variation From a Table

Got It? For the data in the table at the right, does y vary directly with x? If it does, write an equation for the direct variation.

17. Write each ordered pair as the ratio of the y-coordinate to the x-coordinate. Then write the ratio of y to x as a decimal.

(23, 2.25) (1, 20.75) (4, 23)

5 5 5

18. For the data in the table, does y vary directly with x? Yes / No

19. The equation for the direct variation shown is y 5 ? x.



direct variation constant of variation for a direct variation

Rate how well you can work with direct variation.

x y

3

1

4

2.25

0.75

3

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Name Class Date

5-2 Think About a PlanDirect Variation

Electricity Ohm’s Law V 5 I 3 R relates the voltage, current, and resistance of a circuit. V is the voltage measured in volts. I is the current measured in amperes. R is the resistance measured in ohms. a. Find the voltage of a circuit with a current of 24 amperes and a resistance of

2 ohms. b. Find the resistance of a circuit with a current of 24 amperes and a voltage of

18 volts.


1. Does Ohm’s Law represent a direct variation? Explain.

2. If the formula is rearranged to solve for R or I, is it still a direct variation? Explain.


3. For part (a), does Ohm’s Law need to be rearranged to answer the question? Explain. If it does, how should the formula be rearranged?

4. For part (b), does Ohm’s Law need to be rearranged to answer the question? Explain. If it does, how should the formula be rearranged?

Getting an Answer

5. For part (a), substitute the given values into the formula and simplify.

6. For part (b), substitute the given values into the formula and simplify.

Chapter 5 298

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5-2 Practice Form K

Direct Variation

Determine whether each equation represents a direct variation. If it does, fi nd the constant of variation.

1. 3y 1 2 5 2x 2. 2x 2 5y 5 0

3. 27x 5 256y 4. 22 1 4y 1 2 5 8x

Suppose y varies directly with x. Write a direct variation equation that relates x and y. Th en fi nd the value of y when x 5 8.

5. y 5 4 when x 5 8 6. y 5 15 when x 5 5

7. y 5 3 when x 5 8 8. y 5 7.92 when x 5 2.2

Graph each direct variation equation.

9. y 5 3x 10. y 5 2x 11. y 5 23 x

12. Th e perimeter of a square varies directly with the length of one side. What is an equation that relates the perimeter p and length l of the side? What is the graph of the equation?

299 Lesson 5-2

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Name Class Date


Direct Variation

For the data in each table, tell whether y varies directly with x. If it does, write an equation for the direct variation.

13. 14.

Write a direct variation equation that relates x and y. Th en graph the equation.

15. y 5 221 when x 5 7 16. y 5 152 when x 5 25

Tell whether the two quantities vary directly. Explain your reasoning.

17. Sara makes $3.50 more per hour than Pasco.

18. Th e cafeteria provides three meals per day.

19. Jasmine scores 10 points per game.

20. Reasoning How can you tell, by examining the graph, if a line represents a direct variation?

4

2

2

5.4

2.7

2.7

x y6

10

7

6.9

11.5

8.05

x y

Chapter 5 300

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5-2 Standardized Test PrepDirect Variation

Gridded Response

Solve each exercise and enter your answer on the grid provided.

1. Suppose y varies directly with x and y 5 14 when x 5 24. What is the value of y when x 5 26?

2. Suppose y varies directly with x and y 5 25 when x 5 140. What is the value of x when y 5 36?

3. Th e point (12, 9) is included in a direct variation. What is the constant of variation?

4. Th e equation of the line on the graph at the right is a direct variation equation. What is the constant of variation?

5. Th e distance d a train travels varies directly with the amount of time t that has elapsed since departure. If the train travels 475 miles in 9.5 hours, how many miles did the train travel after 4 hours?

1. 2. 3. 4. 5.

xO

y4

2

2

4

2

4 42

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10

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210

9876543210

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3

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2

9876543

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9876543210

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987654321

9876543210

2

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9876543210

987

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987654321

9876543210

2

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987

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2

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301 Lesson 5-2

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Vocabulary

Review

Chapter 5 302

Slope-Intercept Form5-3PART 1

1. Multiple Choice Which equation is NOT a linear equation?

y 5 23x 1 4 y 5 x y 5x52 7 y 5 5x

2. Place a ✓ in the box if the statement applies to the graph of a linear equation. Place an ✗ if it does NOT apply to the graph of a linear equation.

The graph of a linear equation is always a horizontal line.

The graph of a linear equation is always a straight line.

The graph of a linear equation may be shaped like a “U.”

Vocabulary Builder

intercept (noun) IN tur sept

Other Word Forms: intercepted (verb), interception (noun)

Definition: An intercept is a point where someone or something is stopped along its way from one place to another.

Main Idea: You can find the intercept(s) of a graph by finding the point(s) where the graph crosses a coordinate axis.

Related Words: x-intercept; y-intercept

Use Your Vocabulary

Choose the correct word from the list to complete each sentence.

intercept intercepted interception

3. During a football game, the home team’s quarterback threw an 9.

4. The y-coordinate of a point where a graph crosses the y-axis is a y- 9.

5. The teacher 9 the message Charlie was passing to his friend.

A y-intercept is the y-coordinate of a

point where a graph crosses the y-axis.

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d.Key Concept Slope-Intercept Form of a Linear Equation

y 4x 6

Problem 3

Problem 2

x

y

O2 1

1

2

303 Lesson 5-3, Part 1

Writing an Equation in Slope-Intercept Form

Got It? What is an equation of the line with slope 32 and y-intercept 21?

7. Write the numbers 32 and 21 in the correct boxes below.

y 5 m ? x 1 b

y 5 ? x 1

8. An equation in slope-intercept form is .

Writing an Equation From a Graph

Got It? What is an equation of the line shown at the right?

9. Choose two points on the graph to use to find the slope of the line. What two points will you use?

( , ) and ( , )

10. Use the points to find the slope of the line. 11. The slope of the line is .

12. Use the graph to find the y-intercept.

The y-intercept is .

13. Write the slope-intercept form of the equation.

Got It? Reasoning If you use two different points to find the slope of the line, will the equation of the line change? Explain.

14. Now choose two points on the line in the above graph that are different from the ones you chose in Exercise 9. Use these points to calculate the slope of the line.

Th e slope-intercept form of a linear equation of a nonvertical line is y 5 mx 1 b.

Th e slope of the line is m. Th e y-intercept is b.

6. Use the words slope, y-intercept, and slope-intercept form to complete the diagram at the right.

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Lesson Check

Problem 4

Chapter 5 304

Vocabulary Is y 5 5 a linear equation? Explain.

20. Does y 5 5 have a slope? Explain.

__________________________________________________________________________________

21. Find three points that satisfy y 5 5.

( , ) ( , ) ( , )

22. Is y 5 5 a linear equation? Explain.

__________________________________________________________________________________

__________________________________________________________________________________


Writing an Equation From Two Points

Got It? What equation in slope-intercept form represents the line that passes through the points (3, 22) and (1, 23)?

16. Circle the first step to solve this problem. Underline the second step.

Solve for b. Find the slope. Write the slope-intercept form.

17. Use the points (3, 22) and (1, 23) to find the slope of the line.

m 52 (23)

3 25

1

18. Next, find the y-intercept. Substitute the slope for m and the coordinates of one of the points for x and y. Then solve for b.

y 5 m ? x 1 b

5 ? 1 b

5 1 b

5 b

19. Write the equation of the line in slope-intercept form. Substitute the slope for m and the y-intercept for b.

y 5 ? x 1

15. If you use two different points to find the slope of the line, will the equation of the line change? Explain.

__________________________________________________________________________________

__________________________________________________________________________________

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Vocabulary

5-3PART 2

Slope-Intercept Form

Review


Underline the true statement in each pair of statements about the graph of the equation y 5 25x 1 6.

1. The graph is a straight line. The graph is a circle.

2. The slope is 25. The slope is 6.

3. The y-intercept is 26. The y-intercept is 6.

4. The graph has two x-intercepts. The graph has one x-intercept.

Vocabulary Builder

model (noun) MAH dul

Definition: A model is a representation, example, or imitation of a thing or person. Many times a model is smaller than the actual item.

Math Usage: A mathematical model represents a real-life process or a situation through the use of variables and mathematical operations.

Examples: Toy stores sell kits that can be put together to be miniature models of actual airplanes, cars, and ships.

Henrietta earns $12 an hour as a barista in a coffee shop. The equation y 5 12x models how much she makes after working x hours.

Use Your Vocabulary

Draw a line from each situation in Column A to its mathematical model in Column B.

Column A Column B

5. Michael mows x lawns at a rate of $6 an hour. 6 4 x

6. Tessa has x dollars saved and then spends 6x$6 at the store.

7. Georgina walks the same distance each day to x 2 6work. She walks a total of 6 miles over x days.

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Problem 5

x

y

O 54321

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2

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O 4 6 8 102

2

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46810

Problem 6

Chapter 5 306

Graphing a Linear Equation

Got It? What is the graph of y 5 23x 1 4?

8. The ordered pair for the y-intercept, 4, is ( , ).

9. Explain how you will use the slope to find another point on the line.

______________________________________________________________________________________

10. Use the slope, 23, to find the coordinates 11. Use the points you found in Exercises 8 and 10. of another point on the line. What is the graph of y 5 23x 1 4?

Got It? What is the graph of y 5 4x 2 8?

12. The y-intercept of the line is , so

the point ( , ) is on the line.

13. The coordinates of another point

on the line are ( , ).

14. Graph y 5 4x 2 8 on the grid at the right.

Modeling a Function

Got It? A plumber charges a $65 fee for a repair plus $35 per hour. Write an equation to model the total cost y of a repair that takes x hours. What graph models the total cost?

15. Let x 5 the number of hours the plumber works. Let y 5 the total cost of a repair.

When x 5 0, y 5 . So the y-intercept is .

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Hours (x)

Total Cost (y) 65

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Math Success

Lesson Check


16. The slope is the amount of change each hour.

So, the slope is .

17. Write an equation to model the cost of a repair.

18. Complete the table for your equation.

19. Graph the data from the table to model the total cost. Be sure to label the axes.


linear function y-intercept slope-intercept form

Rate how well you can fi nd the slope-intercept form of a linear equation.

Writing Describe two different methods you can use to graph the equation y 5 2x 1 4. Which method do you prefer? Explain.

20. Circle two methods you can use to graph y 5 2x 1 4 correctly.

21. Which method do you prefer? Explain.

____________________________________________________________________________________

____________________________________________________________________________________


Use the y-intercept 2 to plot Choose two values for x Use the y-intercept 4 to plotthe point (0, 2). Th en use the and fi nd the corresponding the point (0, 4).Th en use theslope of 4 to to plot another y-values. Th en plot the slope of 2 to plot anotherpoint. Th en graph the line. points and graph the line. point. Th en graph the line.

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5-3 Think About a PlanSlope-Intercept Form

Hobbies Suppose you are doing a 5000-piece puzzle. You have already placed 175 pieces. Every minute you place 10 more pieces. a. Write an equation in slope-intercept form to model the number of pieces

placed. Graph the equation. b. After 50 more minutes, how many pieces will you have placed?


1. Is this relationship linear? How do you know?


2. How many pieces have you already placed? What does this represent in the slope-intercept form?

3. What two quantities are used to fi nd the rate of change or slope? What is the slope of this relationship?

Getting an Answer

4. Use your answers in Steps 2 and 3 to write an equation in slope-intercept form to model the number of pieces placed.

5. Graph the equation on a coordinate grid.

6. How many pieces will you have placed after 50 more minutes?

Chapter 5 308

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5-3 Practice Form K


Find the slope and y-intercept of the graph of each equation.

1. y 5 22x 1 7 2. y 5 6x 1 11

3. y 5 27x 2 8 4. y 5 22.5x 1 3.2

5. y 5 29 6. y 5 14 x 2 2

7

Write an equation of a line with the given slope m and y-intercept b.

7. m 5 25, b 5 26 8. m 5 1, b 5 24

9. m 5 0.4, b 5 29 10. m 5 0, b 5 3

Write an equation in slope-intercept form of each line.

11. 12.

Write an equation in slope-intercept form of the line that passes through the given points.

13. (21, 2) and (0, 0) 14. (22, 9) and (1, 6)

15. (12, 10) and (16, 8) 16. (24, 21) and (28, 7)

x

y

O2

2

2

2

x

y

O2

2

2

2

309 Lesson 5-3

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Graph each equation.

17. y 5 x 2 2 18. y 5 3x 1 1

19. y 5 2x 2 1 20. y 5 23x 2 2

21. y 5 12 x 1 2 22. y 5 2

45 x 2 5

23. A car is traveling at 45 mi/h. Write an equation that models the total distance d traveled after h hours. What is the graph of the equation?

Chapter 5 310

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5-3 Standardized Test PrepSlope-Intercept Form

Multiple Choice


1. What is an equation of the line shown in the graph at the right?

A. y 5 2 32 x 1 4 C. y 5 2

23 x 1 4

B. y 5 23 x 1 4 D. y 5 2

23 x 1 6

2. What is an equation of the line that has slope 24 and passes through the point (22, 25)?

F. y 5 24x 2 8 G. y 5 24x 2 13 H. y 5 24x 2 5 I. y 5 24x 1 3

3. What is an equation of the line that passes through the points (24, 3) and (21, 6)?

A. y 5 2x 2 7 B. y 5 2x 2 1 C. y 5 7x 1 1 D. y 5 x 1 7

4. Th e data shown in the table is linear. Which equation models the data?

F. y 5 12 x 1 12 H. y 5 2x 1 9

G. y 5 12 x 1 6 I. y 5 2x 2 3

5. Karissa earns $200 per week plus $25 per item she sells. Which equation models the relationship between her pay p per week and the number of items n she sells?

A. p 5 200n 1 25 C. n 5 25p 1 200 B. p 5 25n 1 200 D. n 5 200p 1 25

Short Response

6. What is an equation of the line that passes through (28, 2) and has

slope 2 34? What is the graph of the equation?

y6

4

2

2

xO 24 42

x y

26

10

13

17

15

311 Lesson 5-3

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Vocabulary

Review

Chapter 5 312

Point-Slope Form

1. Circle the equation that has a y-intercept of 3.

y 5 3x 1 4 y 5 4x 2 3 y 5 5x 1 3 y 5 23x 1 2

2. Circle the equation that is in slope-intercept form.

2x 2 y 5 10 x 1 3y 1 11 5 0 y 2 4 5 23(x 1 7) y 5 2x 1 6

3. Circle the statement that is true about the y-intercept of any graph.

occurs where y 5 0 occurs where x 5 0 occurs where graph on the graph on the graph touches the x-axis

Vocabulary Builder

function (noun) FUNGK shun

Related Words: input, output, function rule

Definition: A function is a relationship that assigns exactly one output value to each input value.

Main Idea: A function is used to describe how one value depends on another.

Example: The function machine above shows that the function assigns an output to every input according to a specified rule.

Use Your Vocabulary

Complete each sentence with the appropriate word from the list.

price sun time

4. The length of a shadow is a function of the angle of the 9.

5. The amount of water that has leaked from a leaky faucet is a function of 9.

6. The amount of sales tax you pay is a function of the item’s 9.

InputFunction Rule

Output

5-4PART 1

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Problem 1

Th e point-slope form of an equation of a nonvertical line with slope m and through point (x1, y1) is y 2 y1 5 m(x 2 x1).

7. In the above, what does (x1, y1) represent? 8. What does m represent?

Key Concept Point-Slope Form of a Linear Equation

Problem 2

x

y

O 108642

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4

6

8

246810

4

2

6

8


Writing an Equation in Point-Slope Form

Got It? A line passes through (8, 24) and has slope 23. What is an equation in point-slope form of the line?

9. Use the point-slope form of an equation. For a line that passes through (8, 24) and has slope 23, circle x1 and underline y1.

24 23 8 12

10. Now substitute into point-slope form.

11. An equation of the line is .

Graphing Using Point-Slope Form

Got It? What is the graph of the equation y 1 7 5 245 (x 2 4)?

12. Circle the ordered pair of a point on the graph of y 1 7 5 245 (x 2 4).

(7, 4) (4, 7) (24, 27) (4, 27)

13. Circle the correct description of the slope.

Go up 4 units and left 5 units Go down 4 units and left 5 units Go up 4 units and right 5 units

14. Use your answers to Exercises 12 and 13 to graph the line.

y y1 m (x x1)

y (x )

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Lesson Check

Lesson Check

xO 42

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x

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• Do you know HOW?

Chapter 5 314

What is the slope and one point on the graph of y 2 12 5 49 (x 1 7)?

15. The point-slope form of the equation of a line is .

16. The slope of the line is .

17. The value of y1 is . The value of x1 is .

18. So one point on the line is ( , ).

• Do you know HOW?

What is the graph of the equation y 2 4 5 3(x 1 2)?

19. Use the point-slope form, y 2 y1 5 m(x 2 x1), to complete.

m 5 x1 5 y1 5

20. So, the point (x1, y1) is ( , ).

21. How will you use the slope to find the coordinates of a second point on the line?

_______________________________________________________________________

_______________________________________________________________________

22. Circle the graph of y 2 4 5 3(x 1 2).

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Vocabulary

5-4PART 2

Point-Slope Form

Review


1. Circle the solution of the equation 24x 5 16.

4 24 12 20

2. Circle the solution of the equation 2x 1 1 5 13.

7 6 26 5

3. Circle the equation that has a slope of 4.

y 5 4x 1 3 y 5 24x 2 1 y 5 14x 1 4

Vocabulary Builder

parent function (noun) PEHR unt FUNGK shun

Related Word: family of functions

Definition: A parent function is the simplest function of a group of functions with common characteristics, called a family of functions.

Example: For g(x) 5 x 1 2, h(x) 5 13x, and k(x) 5 2x 1 9, the parent function is

f (x) 5 x.

Use Your Vocabulary

Write the parent function, p(x), for each function f (x).

4. f (x) 5 x 1 5 5. f (x) 5 |x| 2 2 6. f (x) 5 7x

7. f (x) 5 2|x| 8. f (x) 5 28x 9. f (x) 5 2x 1 4

parent functionf(x) x

family of functions

g(x) x 2 k(x) x 9h(x) 13

x

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Problem 3

Problem 4

y

xO 2 4

2

4

2

4

24

(1, 4)

( 2, 3)

Think Write

I can use any two points from the table to find

the slope.

Then I can substitute one point and the slope

into the point-slope equation.

Finally, I can tell what the slope represents.

4570

3 m

y y1 m (x x1)

y (x )

The slope represents a rate of ? .

Chapter 5 316

Using Two Points to Write an Equation

Got It? Use the point (22, 23) to write an equation of the line shown.

10. Follow the steps to write the equation of the line shown.

Using a Table to Write an Equation

Got It? The table shows the number of gallons of water y in a tank after x hours. The relationship is linear. What is an equation in point-slope form that models the data? What does the slope represent?

11. Complete the reasoning model below.

1

2

Use the slope and the point ( 2, 3) to write

an equation of the line in point-slope form.

Find the slope of the line. Use two points and they2 y1x2 x1

slope formula, m = .

y y1 m(x x1)

y (x )

An equation of the line is .

Time, x(h)

Water, y(gal)

3320

4570

7070

10,820

2

3

5

8

Volume of Water in Tank

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Lesson Check


Got It? Reasoning Write the equation from Exercise 11 in slope-intercept form. What does the y-intercept represent?

12. Circle the first step in writing the equation in slope-intercept form.

Add 7070 to each side. Distribute 1250. Add 5 to each side.

13. Write the equation in slope-intercept form.

14. What does the y-intercept in your answer to Exercise 13 represent?

_______________________________________________________________________

_______________________________________________________________________


point-slope form equation

Rate how well you can write equations in point-slope form.


Reasoning Can any equation in point-slope form also be written in slope-intercept form? Give an example to explain.

15. Use point-slope form, y 2 y1 5 m(x 2 x1), and any point and slope to write an equation in point-slope form.

16. Now write your equation in slope-intercept form.

17. Can any equation in point-slope form also be written in slope-intercept form? Yes / No

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5-4 Think About a PlanPoint-Slope Form

Boiling Point Th e relationship between altitude and the boiling point of water is linear. When the altitude is 8000 ft, water boils at 197.6°F. When the altitude is 4500 ft, water boils at 203.9°F. Write an equation giving the boiling point b of water (in degrees Fahrenheit) in terms of the altitude a (in feet). What is the boiling point of water at 2500 ft?


1. What are you given?

2. In general, how can this information be used to answer the question?


3. What is the slope formula?

4. Substitute given values into the slope formula and simplify. Show your work.

5. Which point can be used to write the point-slope form of the equation?

6. What strategy can you use to solve this problem?

7. How can you determine the boiling point of water at 2500 ft?

Getting an Answer

8. Write an equation giving the boiling point b of water (in degrees Fahrenheit) in terms of the altitude a.

9. What is the boiling point of water at 2500 ft? Show your work.

Chapter 5 318

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5-4 Practice Form K

Point-Slope Form

Write an equation in point-slope form of the line that passes through the given point and has the given slope.

1. (1, 3); m 5 5 2. (22, 21); m 5 23 3. (4, 27); m 5 2 14


4. y 1 1 5 3(x 2 2) 5. y 2 4 5 21(x 1 2) 6. y 2 3 5 22(x 1 4)

Graph the line that passes through the given point and has the given slope m.

7. (21, 23); m 5 2 8. (23, 22); m 5 24 9. (22, 6); m 5 2 12

10. Open-Ended Write an equation in each of the following forms that has a

slope of 2 23.

a. point-slope form b. slope-intercept form

11. Writing Describe what you know about the graph of a line represented by the equation y 1 4 5 25(x 2 1).

319 Lesson 5-4

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Point-Slope Form

Model the data in each table with a linear equation in slope-intercept form. What do the slope and y-intercept represent?

12. 13.

Write an equation in point-slope form of each line.

14. 15.

Write an equation in point-slope form of the line that passes through the given points. Th en write the equation in slope-intercept form.

16. (5, 1), (0, 2) 17. (22, 23), (4, 3)

18. (23, 22), (2, 3) 19. (2, 5), (8, 27)

20. Writing Describe how you would use the point-slope form to write the equation of a line that passes through the points (2, 3) and (21, 6) in slope-intercept form.

21. A restaurant’s goal is to serve 600 customers in 8 hours and 900 customers in 12 hours. Write an equation in point-slope form that represents the number of customers served per hour. What is the graph of the equation?

HoursWorked

MoneyEarned ($)

6

4

11

15

49

73.50

134.75

183.75

x

y

O2

2

2

2

TimeRunning (min)

Distance(mi)

40

20

60

100

2

4

6

10

x

y

O2

2

2

2

Chapter 5 320

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5-4 Standardized Test PrepPoint-Slope Form

Multiple Choice


1. Which equation is equivalent to y 2 6 5 212(x 1 4)? A. y 5 26x 2 48 C. y 5 212x 2 42 B. y 5 6x 2 48 D. y 5 212x 2 54

2. Which point is located on the line represented by the equation y 1 4 5 25(x 2 3)?

F. (24, 25) G. (25, 24) H. (3, 24) I. (23, 4)

3. Which equation represents the line that passes through the points (6, 23)and (24, 29)?

A. y 1 4 5 235(x 1 9) C. y 2 3 5 3

5(x 1 6)

B. y 1 4 5 53(x 1 9) D. y 1 3 5 3

5(x 2 6)

4. Which equation represents the line shown in the graph? F. y 5 23x 2 2 G. y 5 3x 1 2 H. y 1 4 5 23(x 2 2) I. y 1 8 5 23(x 2 2)

5. Th e population of a city increases by 4000 people each year. In 2025, the population is projected to be 450,000 people. What is an equation that gives the city’s population p (in thousands of people) x years after 2010?

A. p 5 4x 1 450 C. p 2 15 5 4(x 2 450)

B. p 2 450 5 4(x 2 5) D. p 5 4x 1 15

Short Response

6. Th e table shows the cost of a large cheese pizza with additional toppings on it.

a. What is an equation in point-slope form that represents the relationship between the number of toppings and the cost of the pizza?

b. What is the graph of the equation?

Cost ($)Toppings

235

10.5011.7514.25

xO

y8

4

4

8

4

8 4 8

321 Lesson 5-4

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Review

p

z

Chapter 5 322

5-5 Standard Form

Underline the correct word to complete each sentence.

1. Line z is a horizontal / vertical line.

2. Line p is a horizontal / vertical line.

3. A line with a slope of 0 is horizontal / vertical .

4. A line with an undefined slope is horizontal / vertical .

Vocabulary Builder

standard (adjective) STAN durd

Other Word Forms: standards (plural noun), standardized (adjective)

Main Idea: Something that is standard is well known and widely used.

Example: The standard measure of weight used in the U.S. is the pound.

Math Usage: The standard form of a linear equation is Ax 1 By 5 C , where A, B, and C are real numbers, and A and B are not both zero.

Opposites: different, irregular

Use Your Vocabulary

Underline the correct word(s) to complete each sentence.

5. In gymnastics, judges use a set of standards / standardized to award a score.

6. Most English words have a standard / standardized pronunciation.

7. Many states use standard / standardized tests to assess their students’ performance.

8. Multiple Choice Which linear equation is in standard form?

y 5 26x 1 4 3x 2 7y 5 42

y 5 27x 2 3 y 2 6 5 2(x 1 7)

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Problem 2

Problem 1

323 Lesson 5-5

Finding x- and y-Intercepts

Got It? What are the x- and y-intercepts of the graph of 5x 2 6y 5 60?

Complete each sentence.

9. To find the x-intercept, let y 5 . 10. To find the y-intercept, let x 5 .

11. Find the x-intercept. 12. Find the y-intercept.

5x 2 6 ? 5 60 5 ? 2 6y 5 60

5x 2 5 60 2 6y 5 60

5 60 5 60

5605 5

6026

x 5 y 5

Got It? What are the x- and y-intercepts of the graph of 3x 1 4y 5 12?

13. Find the x-intercept. 14. Find the y-intercept.

3x 1 4 ? 5 12 3 ? 1 4y 5 12

5 12 5 12

5 12 5 12

x 5 y 5

Graphing a Line Using Intercepts

Got It? What is the graph of 2x 1 5y 5 20?

15. Circle the x-intercept of 2x 1 5y 5 20.

x 5 1 x 5 10 x 5 20

16. Circle the y-intercept of 2x 1 5y 5 20.

y 5 25 y 5 24 y 5 4

17. Use the intercepts to graph the line 2x 1 5y 5 20.

x

y

O 10 128642

2

4

6

24681012

4

2

6

5 26

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Problem 3

Problem 4

Problem 5

Chapter 5 324

Graphing Horizontal and Vertical Lines

Got It? What is the graph of the equation x 5 4?

18. The equation x 5 4 means that for all values of y, the value of x is .

19. For the reason given above, the graph of x 5 4 is a horizontal / vertical line.

20. Graph the equation x 5 4.

Transforming to Standard Form

Got It? Write y 2 2 5 213(x 1 6) in standard form using integers.

21. Circle the first step to put y 2 2 5 213 (x 1 6) in standard form.

Solve for y. Multiply both sides by 23. Add x to both sides.

22. Now find the standard form of the equation using integers.

23. The standard form of the equation is ? x 1 ? y 5 0.

Using Standard Form as a Model

Got It? A media download store sells songs for $1 each and movies for $15 each. You have $60 to spend. Write and graph an equation that describes the numbers of songs and movies you can purchase for $60.

24. You cannot buy a fraction of a song or movie. Describe how you will use the graph of the equation to find solutions that make sense.

_______________________________________________________________________

_______________________________________________________________________

x

y

O 53 421

1

2

3

4

5

12345

2

1

3

4

5

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Math Success

Lesson Check

325 Lesson 5-5


linear equation x-intercept standard form

Rate how well you can graph a linear equation using intercepts.

25. Use the model to help you complete the equation.

cost ofa song

Let x the number of songs purchased.

Let y .

Define

Write

Relatenumber of

songs

cost of amovie

isnumberof movies

60

$60

x y

26. Find the intercepts of the equation. 27. Use the intercepts to graph the equation.

x

y

O 5 10 15 20 25 30 35 40 45 50 55 60

1

2

3

4

Num

ber

of M

ovie

s

Number of Songs

Vocabulary Tell whether each linear equation is in slope-intercept form, point-slope form, or standard form.

y 1 5 5 2(x 2 2) y 5 22x 1 5 y 2 10 5 22(x 2 1) 2x 1 4y 5 12

28. Draw a line from each equation in Column A to the form of the equation in Column B.

Column A Column B

y 1 5 5 2(x 2 2) y 5 mx 1 b (Slope-Intercept Form)

y 5 22x 1 5 y 2 y1 5 m(x 2 x1) (Point-Slope Form)

y 2 10 5 22(x 2 1) Ax 1 By 5 C (Standard Form)

2x 1 4y 5 12


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Sports A football team scores 63 points. All of the points come from fi eld goals worth 3 points and touchdowns (with successful extra-point attempts) worth 7 points. Write and graph a linear equation that represents this situation. List every possible combination of fi eld goals and touchdowns the team could have scored.


1. What are you given?

2. How can touchdowns and fi eld goals be represented? How can these be written as terms to represent the point value of each?


3. What is the equation in standard form that models the situation?

4. How can you fi nd the y-intercept?

5. How can you fi nd the x-intercept?

6. How can you use the intercepts to graph the line?

Getting an Answer

7. Graph the relation on a coordinate grid.

8. Use the graph to determine and list all of the combinations.

5-5 Think About a PlanStandard Form

Chapter 5 326

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5-5 Practice Form K

Standard Form

Find the x- and y-intercepts of the graph of each equation.

1. x 1 y 5 23 2. 2x 2 4y 5 28

3. x 1 5y 5 210 4. 23x 1 2y 5 12

Draw a line with the given intercepts.

5. x-intercept: 2 6. x-intercept: 24 y-intercept: 23 y-intercept: 22

Graph each equation using x- and y-intercepts.

7. 3x 1 y 5 22 8. 22x 1 y 5 1 9. x 2 y 5 4

10. 26x 1 y 5 24 11. 2x 2 3y 5 26 12. 6x 1 8y 5 24

For each equation, tell whether its graph is a horizontal or a vertical line.

13. x 5 21 14. y 5 5


15. x 5 25 16. y 5 6

327 Lesson 5-5

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Standard Form

17. Writing Explain how y 2 2 5 2(x 1 6) can be rewritten into standard form. Th en show your work in transforming the equation to standard form.

Write each equation in standard form using integers.

18. y 5 x 1 6 19. y 1 5 5 2(x 1 3)

20. y 2 1 5 2 12 (x 2 4) 21. y 5 2

23 x 1 6

22. You work two jobs. At the fi rst job, you earn $10 per hour. At the second job, you earn $12 per hour. You earned $440 last week. Write and graph an equation that represents this situation. What are three combinations of hours you could have worked at each job?

23. Mike was the kicker for the football team. He scored 56 points during the season kicking fi eld goals (3 points) and extra points (1 point). Write and graph an equation that represents this situation. What are three combinations of fi eld goals x and extra points y he could have made?

For each graph, fi nd the x- and y-intercepts. Th en write an equation in standard form using integers.

24. 25.

Find the x- and y-intercepts of the line that passes through the given points.

26. (2, 22), (6, 24) 27. (21, 23), (4, 2)

x

y

O2

2

2

2

x

y

O4

4

4

4

Chapter 5 328

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Multiple Choice


1. What is y 5 253 x 2 6 written in standard form using integers?

A. 53 x 1 y 5 26 B. 5x 1 3y 5 26 C. 5x 1 3y 5 218 D. 25x 1 3y 5 6

2. Which of the following is an equation of a vertical line? F. 4x 1 5y 5 0 G. 24 5 16x H. 3y 5 29 I. 4x 1 5y 5 21

3. What are the x- and y-intercepts of the graph of 27x 1 4y 5 214 A. x-intercept: 27 C. x-intercept: 22 y-intercept: 4 y-intercept: 3.5

B. x-intercept: 7 D. x-intercept: 2 y-intercept: 24 y-intercept: 23.5

4. Cheryl is planning to spend $75 on a Christmas gift for her father. He needs new socks and ties. A store has socks s and ties t on sale for $4 and $11, respectively. Which equation models this situation?

F. 4s 1 11t 5 75 H. s 5 15t 1 75

G. 11s 1 4t 5 75 I. t 5 4s 2 11

Extended Response

5. Th e grocery store is selling eggs for $2 per dozen and bacon for $5 per pound. You plan to spend $50 in food for the benefi t breakfast. Write and graph an equation that represents this situation. What are three combinations of dozens of eggs and pounds of bacon you can purchase?

5-5 Standardized Test PrepStandard Form

329 Lesson 5-5

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Vocabulary

Review

Chapter 5 330

5-6 Parallel and Perpendicular Lines

1. Circle the product of a number and its reciprocal.

100 1 0 21

2. Circle the pairs of numbers that are reciprocals.

7 and 17 1 and 21 0 and 012 2

34 and 24

3

Vocabulary Builder

parallel (adjective) PA ruh lel

Related Word: perpendicular (adjective)

Math Usage: Lines that are parallel are lines in the same plane that never intersect.

Using symbols: *AB) 6 *CD) means line AB is parallel to line CD.

Example: The stripes on the American flag are parallel.

Use Your Vocabulary

Picture A Picture B Picture C

Complete each sentence with parallel or perpendicular.

3. The railroad tracks in Picture A are 9.

4. The window bars in Picture B that do NOT meet are 9.

5. The roads in Picture C are 9.

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Problem 1

Key Concept Slopes of Parallel Lines

Key Concept Slopes of Perpendicular Lines

x

y

O 42

2

4

24

4

2

x

y

O 42

2

4

24

2

4

331 Lesson 5-6

Writing an Equation of a Parallel Line

Got It? A line passes through (12, 5) and is parallel to the graph of y 5 23x 2 1.

What equation represents the line in slope-intercept form?

7. The slope of the graph of y 5 23x 2 1 is .

8. The slope of any line parallel to the graph of y 5 23x 2 1 is .

9. Use point-slope form to find an equation of the line that passes through (12, 5) and uses the slope from Exercise 8.

y 2 y1 5 m(x 2 x1)

Nonvertical lines are parallel if they have the same slope and diff erent y-intercepts. Vertical lines are parallel if they have diff erent x-intercepts.

6. Draw a line from each equation in Column A to an equation whose graph is parallel in Column B.

Column A Column B

y 5 2x 1 4 x 5 2

y 5 13x 2 2 y 5 2x 2 4

x 5 3 y 5 13x 2 1

Two nonvertical lines are perpendicular if the product of their slopes is 21. Two numbers whose product is 21 are called opposite reciprocals. A vertical line and a horizontal line are also perpendicular.

10. Multiple Choice The slope of a line is 2. What is the slope of a line perpendicular to that line?

2 22

12 21

2

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Problem 3

Slope of the given line opposite reciprocal

(slope of perpendicular line)

1

Problem 2

Chapter 5 332

Classifying Lines

Got It? Are the graphs of the equations y 5 34x 1 7 and 4x 2 3y 5 9 parallel,

perpendicular, or neither? Explain.

11. Write the equation 4x 2 3y 5 9 in slope-intercept form.

12. The slope of the line y 5 34x 1 7 is . 13. The slope of the line 4x 2 3y 5 9 is .

14. Are the lines parallel, perpendicular, or neither? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Writing an Equation of a Perpendicular Line

Got It? A line passes through (1, 8) and is perpendicular to the graph of y 5 2x 1 1. What equation represents the line in slope-intercept form?

15. Find the slope of the perpendicular line.

16. The slope of the perpendicular line is .

17. Use point-slope form and the point (1, 8) to write an equation of the perpendicular line.

y 2 y1 5 m ? (x 2 x1)

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0 2 4 6 8 10

Math Success

Lesson Check

Problem 4

2 4 6 8 10

10

12

12

2

4

6

8

O

y

x

Existingbeam

CEILING VIEW

333 Lesson 5-6


parallel lines perpendicular lines opposite reciprocals

Rate how well you can write equations of parallel and perpendicular lines.


Compare and Contrast How is determining if two lines are parallel similar to determining if they are perpendicular? How are the processes different?

21. To determine if two lines are parallel, 22. To determine if two lines are perpendicular, what do you need to do? what do you need to do?

23. How are the processes similar? 24. How are the processes different?

Solving a Real-World Problem

Got It? An architect uses software to design a ceiling. The architect needs to enter an equation that represents a new beam. The new beam will be parallel to the existing beam, which is shown by the red line. The new beam will pass through the corner at (0, 10). What is an equation in slope-intercept form that represents the new beam?

18. Use the slope formula to find the slope of the red line that represents the existing beam.

19. In order for the new beam to be parallel to the existing beam,

their slopes should be the same / opposite reciprocals .

20. Now find the equation of the line that will be parallelto the existing line and will pass through (0, 10).

m 52

25

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Agriculture Two farmers use combines to harvest corn from their fi elds. One farmer has 600 acres of corn, and the other has 1000 acres of corn. Each farmer’s combine can harvest 100 acres per day. Write two equations for the number of acres y of corn not harvested after x days. Are the graphs of the equations parallel, perpendicular, or neither? How do you know?


1. What is the diff erence between the two farms? What is the same?

2. How can you determine if the graphs of two equations are parallel, perpendicular, or neither?


3. What is an algebraic expression that represents the amount of corn each farmer can harvest per day?

4. Write an equation representing the number of acres y of corn not harvested after x days on the farm with 600 acres.

5. Write an equation representing the number of acres y of corn not harvested after x days on the farm with 1000 acres.

Getting an Answer

6. Write the equations in Steps 4 and 5 in slope-intercept form.

7. What are the slopes of the equations?

8. Are the graphs of the equations parallel, perpendicular, or neither? Explain.

5-6 Think About a PlanParallel and Perpendicular Lines

Chapter 5 334

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5-6 Practice Form K

Parallel and Perpendicular Lines

Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of the given equation.

1. (21, 3); y 5 2x 2 8 2. (2, 6); y 5 23x 1 5

3. (23, 12); y 5 2 13 x 1 7 4. (8, 210); y 5 3

4 x 1 1

Determine whether the graphs of the given equations are parallel, perpendicular, or neither. Explain.

5. y 5 25x 1 9

5x 1 y 5 221 6. x 5 1

10

y 5 110

7. y 5 24x 1 14

2x 1 4y 5 14 8. y 5 6

7 x 1 4

y 5 2 67 x 2 5

Determine whether each statement is always, sometimes, or never true. Explain.

9. Two lines with diff erent slopes are parallel.

10. Two lines with the same y-intercept are perpendicular.

11. Two lines whose slopes are opposites of each other are perpendicular.

335 Lesson 5-6

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Parallel and Perpendicular Lines

Write an equation of the line that passes through the given point and is perpendicular to the graph of the given equation.

12. (6, 22); y 5 23x 1 4 13. (2, 7); y 5 12 x 2 11

14. (25, 26); x 1 y 5 6 15. (4, 25); 2x 1 2y 5 6

16. Open-Ended Write the equations of three lines whose graphs are parallel to y 5 2x 1 11.

17. Open-Ended Write the equations of two lines whose graphs are

perpendicular to y 5 2 13 x 2 9.

18. What is the slope of a line that is parallel to y 5 2?

19. What is the slope of a line that is perpendicular to y 5 2?

20. What is the slope of a line that is parallel to x 5 24?

21. What is the slope of a line that is perpendicular to x 5 24?

22. On a map, Center St. passes through coordinates (5, 23) and (3, 7). Merrie Rd. intersects Center St. and passes through coordinates (2, 6) and (23, 5). Are these streets perpendicular? Explain.

Chapter 5 336

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Multiple Choice


1. Which equation has a graph parallel to the graph of 9x 1 3y 5 222?

A. y 5 3x 2 22 B. y 5 23x 1 8 C. y 5 13 x 1 12 D. y 5 2

13 x 2 2

2. Which equation has a graph perpendicular to the graph of 7x 5 14y 2 8?

F. y 5 22x 2 7 G. y 5 2 12 x 1 4 H. y 5 1

2 x 2 1 I. y 5 2x 1 9

3. Which equation is the equation of a line that passes through (210, 3) and is perpendicular to y 5 5x 2 7?

A. y 5 5x 1 53 B. y 5 2 15 x 2 7 C. y 5 2

15 x 1 1 D. y 5 1

5 x 1 5

4. Which of the following coordinates for P will make *

MN)

parallel to *

OP)

in the diagram at the right? F. (22, 25) H. (3, 2) G. (23, 6) I. (3, 5)

5. Segment XY represents the path of an airplane that passes through the coordinates (2, 1) and (4, 5). What is the slope of a line that represents the path of another airplane that is traveling parallel to the fi rst airplane?

A. 22 C. 12

B. 2 12 D. 2

Short Response

6. A city designer is drawing the road map for a new housing development. Palm St. runs through the coordinates (11, 5) and (21, 1) on the map. Pepperdine St. is going to run perpendicular to Palm St. Th e coordinates of Pepperdine St. are (4, 7) and (7, y). What is the value of y? What is the equation for the line representing Pepperdine St. in slope-intercept form?

5-6 Standardized Test PrepParallel and Perpendicular Lines

xO

y8

4

4

8

4

8 4 8

N(4, 7)

M(22, 25)

O(23, 5)

337 Lesson 5-6

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x

y

O 54321

1

2

3

4

12345

2

1

3

4

0 x

y

0 x

y

0 x

y

Chapter 5 338

5-7 Scatter Plots and Trend Lines

A scatter plot is a graph that relates two sets of data. Plot each ordered pair on the graph at the right to make a scatter plot.

1. (2, 3)

2. (21, 22)

3. (0, 2)

4. (22, 0)

Vocabulary Builder

correlation (noun) kawr uh LAY shun

Related Words: correlate (verb), relationship (noun), relate (verb), scatter plot (noun)

Definition: A correlation is a measure of the strength of a relationship between two quantities.

Example: The more a student studies, the higher the student’s grades tend to be. So, there is a correlation between time spent studying and grades.

Use Your Vocabulary

Label each scatter plot positive correlation, negative correlation, or no correlation.

5. 6. 7.

y increases as x increases y decreases as x increases x and y are not related

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d.Problem 1

x

y

O 1 2 3 4 5 6 7 8 9 10 11 12 151413

0.5

1

1.52

2.5

3

3.5

4

4.5

5

Dollars Spent

Dollars Spent

Gallons Bought

10

2.5

11

2.8

9

2.3

10

2.6

13

3.3

5

1.3

8

2.2

4

1.1

Gasoline Purchases

339 Lesson 5-7

Making a Scatter Plot and Describing Its Correlation

Got It? Make a scatter plot of the data in the table. What type of relationship does the scatter plot show?

8. Let x 5 dollars spent.

Let y 5

.

9. Use the data to make a scatter plot.

10. Underline the correct word or words to complete each sentence.

The number of gallons bought

tends to increase / decrease

as the number of dollars spent

increases / decreases.

The two sets of data have a

positive / negative correlation.

Got It? Reasoning Consider the population of a city and the number of letters in the name of the city. Would you expect a positive correlation, a negative correlation, or no correlation between the two sets of data? Explain your reasoning.

11. As an example, think of the city or town that you live in. How many letters are in the name of your city and approximately how many people live there?

_______________________________________________________________________

12. Now think of another city of a very different size than the one you chose for Exercise 11. How many letters are in the name of this city and approximately how many people live there?

_______________________________________________________________________

13. Is the size of either city dependent on the number of letters in its name? Yes / No

14. What kind of correlation would you expect between the two sets of data? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

A trend line is a line on a scatter plot, drawn near the points, that shows a correlation. Th ere should be about the same number of points above the line as below it.

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Problem 3

Problem 2

Age (month)

Body Length (in.)

1

8.0

2

11.75

3

15.5

4

16.7

5

20.1

6

22.2

8

26.5

9

29.0

Body Length of a Panda

x

y

O 1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

Body

Len

gth

(in.)

Age (month)

Chapter 5 340

Writing an Equation of a Trend Line

Got It? Make a scatter plot of the data. Draw a trend line and write its equation. What is the approximate body length of a 7-month-old panda?

15. Make a scatter plot and draw a trend line. 16. Write the equation of the trend line that you drew.

17. Use the equation of your trend line to estimate the body length of a 7-month-old panda.

18. A 7-month-old panda would be approximately inches in length.

Finding the Line of Best Fit

Got It? For data of tuition and fees charged at public four-year colleges, the equation of the line of best fit is y 5 409.43x 2 815,446.71, where x 5 the year at the beginning of the academic year and y 5 cost. Predict the cost of attending a public four-year college in the 2016–2017 academic year.

19. Let x 5 .

20. Complete the steps to find the estimated cost.

y 5 409.43 ? 2 815,446.71

y 5 2 815,446.71

y <

21. The cost of attending a public four-year college in the 2016–2017 academic year

will be about $ .

Causation is when a change in one quantity causes a change in a second quantity. A correlation between quantities does not always imply causation.

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Lesson Check

Problem 4

x

y

O 10987654321

2

4

8

8

4

2

6

8

10

10

x

y5

2

4

4

1

7

0

9

10

1

7

0

341 Lesson 5-7


scatter plot correlation trend line causation

Rate how well you can make a scatter plot and determine the type of correlation.


Error Analysis Refer to the table below. A student says that the data have a negative correlation because as x decreases, y also decreases. What is the student’s error?

24. Make a scatter plot of the data.

25. The scatter plot shows a positive / negative correlation.

26. Explain the student’s error.

__________________________________________________________________________________

__________________________________________________________________________________

Identifying Whether Relationships Are Causal

Got It? Consider the cost of a family's vacation and the size of their house. Is there likely to be a correlation? If so, does the correlation reflect a causal relationship? Explain.

22. Is there likely to be a correlation between the cost of a family’s vacation and the size of their house? Explain.

__________________________________________________________________________________

23. If there is a correlation, does the correlation reflect a causal relationship? Explain.

__________________________________________________________________________________

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5-7 Think About a PlanScatter Plots and Trend Lines

U.S. Population Use the data below.

a. Make a scatter plot of the data pairs (male population, female population). b. Draw a trend line and write its equation. c. Use your equation to predict the U.S. female population if the U.S. male

population increases to 150,000,000. d. Reasoning Consider a scatter plot of the data pairs (year, male

population). Would it be reasonable to use this scatter plot to predict the U.S. male population in 2035? Explain your reasoning.

1. Make a scatter plot of the data pairs using the male population for the x-coordinates and the female population for the y-coordinates for each year.

2. Draw the trend line onto the scatter plot.

3. How do you determine the equation of a trend line? What is the equation of this trend line? Show your work.

4. Substitute 150,000,000 for x to predict the female population.

5. Make a scatter plot of the data pairs (year, male population).

6. Would it be reasonable to use this scatter plot to predict the U.S. male population in 2035? Explain your reasoning.

Year 2000 2001 2002 2003 2004 2005 2006

138,482 140,079 141,592 142,937 144,467 145,973 147,512

143,734 145,147 146,533 147,858 149,170 150,533 151,886

Male

Female

Estimated Population of the United States (thousands)

SOURCE: U.S. Census Bureau.

Chapter 5 342

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5-7 Practice Form K

Scatter Plots and Trend Lines

For each table, make a scatter plot of the data. Describe the type of correlation the scatter plot shows.

1. 2.

3. Use the table below and a graphing calculator.

a. Make a scatter plot of the data pairs (years since 1960, population).

b. Draw a line of best fi t for the data.

c. Write an equation for the line of best fi t.

d. According to the data, what will the estimated resident population in Ohio be in 2030?

Tips Earned by Waiter

HoursWorked

Tips ($)

2 3 6 8 9

36 62 120 148 165

Foots Size and Height

FootSize (in.)

Height(in.)

10 13 8 6 11

70 77 66 61 72

Ohio Resident Population

Population(thousands)

SOURCE: U.S. Census Bureau

Year 1960 1970 1980 1990 2010

9706 10,652 10,798 10,847

2000

11,353

2005

11,478 11,576

343 Lesson 5-7

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Scatter Plots and Trend Lines

In each situation, tell whether a correlation is likely. If it is, tell whether the correlation refl ects a causal relationship. Explain your reasoning.

4. the number of cabinets Omar assembles and the amount of time it takes him to assemble one

5. the number of darts thrown at the dart board and Jackie’s average score in the game of darts

6. the dates of the summer beach vacation and the weather

7. Open-Ended Describe a real world situation that would show a strong positive correlation. Explain your reasoning.

8. Writing Describe how extrapolation could be useful in a business application.

9. Use the table below and a graphing calculator.

a. Make a scatter plot of the data pairs (years since 2001, cars sold).

b. Draw a line of best fi t for the data.

c. Write an equation for the line of best fi t.

d. According to the data, about how many hybrid cars will be sold in 2020?

Sales of Hybrid Cars in the U.S.

Cars Sold(thousands)

SOURCE: hybridcars.com

Year 2001 2002 2003 2004 2007

20 38 54 84

2005

206

2006

252 288

Chapter 5 344

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5-7 Standardized Test PrepScatter Plots and Trend Lines

Multiple Choice


1. For the following situation, determine if there is a correlation. If there is a correlation, is it a causal relationship?the number of hours practicing at the batting cages and your batting average

A. negative correlation and a causal relationship B. positive correlation but not a causal relationship C. positive correlation and a causal relationship D. no correlation

2. When evaluating data on a scatter plot, what can be used to make predictions about the future?

F. interpolation H. correlation coeffi cient G. extrapolation I. causation

3. Mr. Bolton has worked for the same company for 17 years. What relationship would you expect between the number of years he has been with the company and his annual salary?

A. positive correlation C. no correlation B. negative correlation D. none of the above

4. A city had a population of 150,000 people in 1990. Th e population growth of the city is represented by the equation p 5 5t 1 150 where p is the population in thousands and t is the time in years since 1990. In what year will the population have doubled?

F. 1993 G. 2000 H. 2020 I. 2030

5. What type of correlation is represented by the data in the scatter plot?

A. positive correlation C. no correlation B. negative correlation D. none of the above

Short Response

6. Use the scatter plot to answer the following questions. a. What is an equation of the trend line for the data?

b. What would the earnings be for 40 hours worked?

20 1 3 5 701020304050607080

4 6 8Hours Worked

Earn

ings

($)

2 3 5 71001020304050607080

4 6 8Number of Siblings

Hei

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(in.)

345 Lesson 5-7

5-1 rate of change and slope - somersetcanyons.com 5... · compare and contrast how does finding a...

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