Name Class Date. Additional Vocabulary Support

(1)

Prentice Hall Algebra 1 • Teaching Resources

Concept List

negative slope positive slope rate of change

rise run slope

slope formula slope of horizontal line slope of vertical line

Choose the concept from the list above that best represents the item in each box.

1. _x^y² ²^y¹

2 2x₁ 2. 3.

4. vertical change horizontal change

5. 6.

7. 8. change in the dependent variable change in the independent variable

9.

5-1 Additional Vocabulary Support

Rate of Change and Slope

y

x O

y

x O

3 2 1 0 54 32 10

y

x O

y

x

O 0 1 2 3

54 32 10 slope formula

slope or rate of change

negative slope rate of change or slope slope of vertical line slope of horizontal line rise

run positive slope

(2)

5-1 Think About a Plan

Rate 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?

change in dependent variable change in independent variable

proﬁ t, time; dollars, months

positive; proﬁ t increase over time

Proﬁ t depends on time, so proﬁ t is dependent and time is independent

r 5 $2050 per month

the slope

rate5change in y change in x

(3)

Prentice Hall Gold Algebra 1 • Teaching Resources

5-1 ^Practice Rate of Change and Slope

^{Form G} Determine whether each rate of change is constant. If it is, fi nd the rate of

change and explain what it represents.

1. 2. 3.

Find the slope of each line.

4. 5. 6.

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

7. (2, 1), (0, 0) 8. (4, 5), (6, 2) 9. (3, 8), (7, 3) 10. (1, 0), (24, 2) ^11.(8, 24), (26, 23) ^12.(22, 23), (6, 5)

13. 14. 15.

Goals Games

Hockey Team’s Offense

1 2 3

2 4 6

Miles Gallons

Miles Per Gallon

1 3 5 7

28 84 140 196

Cars Hours

Cars Washed

1 2 3 4

4 8 12 16

x O

6 y

2 4 2

Ź2

Ź4 Ź2 4

x O

6 y

2 4 2

Ź2

Ź4 Ź2 4

x O

4 y

2 2

Ź4 Ź2

Ź4 Ź2 4

x O

4 y 2

Ź4 Ź2

Ź4 Ź2 2 4

x O

4 y

2 2

Ź4 Ź2

Ź4 Ź2 4 Ź2 ^O ^x

4 y

2 2

Ź4 Ź2

Ź4 4

yes; 2; goals per games played

2

1 2

2²₅

0

yes; 28; miles per gallon

3

2³₂ 2₁₄¹

undeﬁ ned

yes; 4; cars washed per hour

21

2⁵₄ 1

0

(4)

5-1 ^Practice

(continued) Form G

Rate of Change and Slope

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

16. Th e cost of tickets to the amusement park is $19.50 for 1 ticket and $78 for 4 tickets.

17. Th e late fee is $2 regardless of the number of days the movie is late.

18. On the trip, Jerry had his cruise control set at 60 mi/h for 4 hours.

19. Th e contract states that every day past the agreed upon completion date the project is not fi nished, the price is reduced by $25.

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

20. Shelly delivered 12 newspapers after 20 minutes and 36 papers after 60 minutes.

21. Two pounds of apples cost $3.98. Six pounds cost $11.94.

22. An airplane ascended 3000 feet in 10 minutes and 4500 feet in 15 minutes.

23. (25, 0), (25, 5) ^24.(22, 24), (21.5, 21.5) ^25.(4.75, 23.575), (2.25, 1.425)

26. Q2¹₄, ³_4R, _Q¹₂, 2³_4R ^27.Q2

5, ³_7R, _Q¹₅, ⁴_7R 28. (23.35, 6.5), (5.65, 23.5) 29. Writing Explain why the slope of a horizontal line is always zero.

30. Writing Describe how to draw a line that passes through the origin and has a slope of 2²₃.

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

31. (7, 4), (3, y); slope 5¹₄ ^32.(5, y), (6, 4); slope 5 0 33. (x, 5), (23, 6); slope 5 21 ^34.(212, 9), (x, 22); slope 5 2¹₂

positive; 19.5

zero; 0 zero; 0

negative; 225

ind: time; dep: number of papers delivered; 0.6 papers/min ind: weight; dep: cost; $1.99/lb

ind: time; dep: height; 300 ft/min

undeﬁ ned

The change in the dependent variable is 0 and ⁰_a5 0.

3

On a coordinate grid, plot (0, 0). Move down 2 and right 3 and plot the point (3, 22).

Draw a line through the points.

5

4

22 10

2¹⁰₉ 2⁵₇

22 22

(5)

Prentice Hall Foundations Algebra 1 • Teaching Resources

5-1 ^Practice Rate of Change and Slope

^{Form K} Each rate of change is constant. Find the rate of change and explain what it

represents.

1. 2.

3. 4.

5. 6.

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

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

11. 12.

Fences Painted Hours Fences

6 3 9 12

1 2 3 4

x y

O Ź2

Ź2 2 2

x y

O Ź2

Ź2 2 2

x y

O Ź2

Ź2 2 2

Miles Per Hour Hours Miles

4 2 6 8

70 140 210 280

x y

O Ź2

Ź2 2 2

x y

O Ź2

Ź2 2 2

x y

O Ź2

Ź2 2 2 1

2⁴₅ 23

2 1

undeﬁ ned 0

22

2¹₃ 2²₃

0.33 fences 1 hour ;

One-third of a fence is painted each hour.

35 miles

1 hour ; They are travelling at 35 miles per hour.

(6)

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.

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

19. Q13, ²_5R, Q2¹₃, ³_5R ^20.Q2³₄, ²_3R, Q2³₄, ⁵_3R 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 ⁸ ^24.(9, 4), (x, 6); slope 5 2¹₃ ³

undeﬁ ned 22

2₁₀³ undeﬁ ned

positive; ^22.50₁

positive; ^9.50₁

independent: gallons of milk; dependent: cost; rate of change52.95 dollars 1 gallon

independent: minutes; dependent: envelopes stuffed; rate of change510 envelopes 1 minute

The slope is always undeﬁ ned because any two points will have the same x-coordinates which means the run will always be zero. Since the denominator is zero, the slope is undeﬁ ned.

Answers may vary. Sample: Plot a point at the origin. Since the slope is ³₅, move up 3 units and to the right 5 units and plot a point. From this point, go up 3 units and to the right 5 units and plot another point. Draw a line through these 3 points.

(7)

5-1 Standardized Test Prep

Rate 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.2¹₃ ^{D. 1}₃

2. A line has slope 2⁵₃. 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 8 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?

x O

175 y 125 10075 150

2550 (2, 44)

(8, 176)

2 4 6 8 10 1214 Hours Reading

Pages Read

C

I

B

I

B

positive; ₁₅² mi/min or 8 mi/h [2] Both parts answered correctly.

[1] One part answered correctly.

[0] Neither part answered correctly.

(8)

5-1 ^Enrichment Rate of Change and Slope

A wildlife biologist is observing interactions among animals in a forest. She notices a fox sniffi ng around, in search of lunch. She also notices a rabbit happily chewing on some grass about 60 meters away. Th e fox looks up, notices the rabbit and starts heading towards it at a constant rate. Th ere are a number of diff erent possible outcomes regarding this situation and some are modeled in the graphs shown below. Th e fox’s path is shown in black and the rabbit’s is shown in gray.

1. Explain what is happening regarding rate of change and slope of each line.

2. In each graph, what will happen regarding the fox and rabbit? How does the slope assure this outcome?

3. Assume the rabbit was a bit confused and wound up running towards the fox at a rate of 10 meters per second. Sketch the graph. Note the slope.

Determine the approximate time when the fox would catch the rabbit.

0 1 2 3 4 5 6 7 8 9 10 200

150 50 100 0

Time Graph A

Distance

0 1 2 3 4 5 6 7 8 9 10 200

150 50 100 0

Time Graph B

Distance

0 1 2 3 4 5 6 7 8 9 10 200

150 50 100 0

Time Graph C

Distance

0 1 2 3 4 5 6 7 8 9 10 200

150 50 100 0

Time Graph D

Distance

A: Fox and rabbit run at the same rate, 10 m/s, so the slopes are the same.

B: Fox runs at 10 m/s and rabbit runs at 30 m/s, so the rabbit’s graph is steeper.

C: Rabbit remains still, and fox runs at 10 m/s, so the rabbit’s graph is horizontal.

D: Fox runs at 10 m/s and rabbit runs at about 2 m/s, so the fox’s graph is steeper.

In A and B the rabbit gets away; in C and D the fox catches the rabbit. When the slope of the fox’s line is greater than the slope of the rabbit’s line, then the lines intersect at a time t S 0 and the fox catches the rabbit. When the slope of the fox’s line is less than the slope of the rabbit’s line, the lines do not intersect for t S 0 and the rabbit escapes.

x y

2 50 100

4 6 8 10 O

Time (sec)

Distance (m)

Rabbit

Fox

slope: 210, 3 s

(9)

5-1 ^Reteaching Rate of Change and Slope

Th e rate of the vertical change to the horizontal change between two points on a line is called the slope of the line.

slope5 vertical change

horizontal change 5^rise_run Th ere are two special cases for slopes.

• A horizontal line has a slope of 0.

• A vertical line has an undefi ned slope.

Problem

What is the slope of the line?

slope5 vertical change

horizontal change5 ^rise_run 5¹₃

Th e slope of the line is ¹₃.

In general, a line that slants upward from left to right has a positive slope.

Problem

What is the slope of the line?

slope5 vertical change

horizontal change5 ^rise_run 5²²₁

5 22

Th e slope of the line is 22.

In general, a line that slants downward from left to right has a negative slope.

x O

4 y

4 2 2

Ź4 Ź2

Ź2 6

(1, 1) (4, 2) rise 5 1 run 5 3

x O

4 y

2

Ź4 Ź2

Ź4 4

(1, 1) (0, 3)

Ź2^{run 5 1} rise 5 22

(10)

5-1 Reteaching

(continued)

Rate of Change and Slope

Exercises

1. 2. 3.

Suppose one point on a line has the coordinates (x1, y1) and another point on the same line has the coordinates (x2, y2). You can use the following formula to fi nd the slope of the line.

slope5^rise_run 5_x^y²₂ ^{2 y}_{2 x}¹₁, where x₂ 2 x₁2 0

Problem

What is the slope of the line through R(2, 5) and S(21, 7)?

slope5 ^y_x²₂^{2 y}_{2 x}¹₁

5_{21 2 2}⁷^{2 5} ^Let^y²^{5 7 and y}¹^{5 5.}

Let _x₂_{5 21} and _x₁_{5 2}. 5₂₃² 5 2²₃

Exercises

4. (0, 0), (4, 5) 5. (2, 4), (7, 8) 6. (22, 0), (23, 2)

7. (22, 23), (1, 1) ^8.(1, 4), (2,23) ^9.(3, 2), (25, 3)

x O

4 y

4 2 2

Ź4 Ź2

Ź2 6

(1, 0)

(5, 21) ^x

O 6 y

4 2 4 2

Ź2

Ź2 6

(1, 2) (6, 5)

x O

6 y

4 2 4 2

Ź2

Ź2 6

(1, 4) (5, 4)

2¹₄ ³₅ 0

5 4

4 3

4 5

27

22

2¹₈

(11)

There are two sets of note cards below that show how Latoya finds a direct variation equation relating x and y. Suppose y varies directly with x, and y 5 36 when x 5 9. She also wants to find the value of y when x 5 7. The set on the left explains her thinking. The set on the right shows the steps. Write the thinking and the steps in the correct order.

Think Cards Write Cards

Think Write

Start with the function form of a direct variation.

Divide each side by 9 to solve for k.

5-2 Additional Vocabulary Support

Direct Variation

y 5 4x 36 5 k(9) y 5 4(7) 5 28

y 5 kx

4 5 k Find the value of y when x 5 7.

Step 1

Step 2

Step 3

Step 4

Step 5 Substitute 9 for x and 36 for y.

Write an equation. Substitute 4 for k in y 5 kx.

y 5 kx

y 5 4(7) or 28 4 5 k

y 5 4x 36 5 k(9) First, she should start with the

function form of a direct variation.

Second, she should substitute 9 for x and 36 for y.

Next, she should divide each side by 9 to solve for k.

Then, she should write an equation.

Substitute 4 for k in y = kx.

Finally, she should find the value of y when x = 7.

(12)

5-2 Think About a Plan

Direct Variation

Electricity Ohm’s Law V5 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.

Understanding the Problem

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.

Planning the Solution

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.

yes; the ratio of V to R is constant, and the ratio of V to I is constant.

R and I would be in direct variation with V but not with each other.

no; you want to ﬁ nd V

yes; you want to ﬁ nd R; R5^V_I

48 volts

0.75 ohms

(13)

5-2 ^Practice Direct Variation

^{Form G} Determine whether each equation represents a direct variation. If it does, fi nd

the constant of variation.

1. 28y 5 2x ^2.3x 1 4y 5 25 ^3.12x 5 236y

4. 27 1 9y 1 7 5 2x ^5.y 2 12 5 12x ^6.5x 1 12.5y 5 0

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 x5 8.

7. y 5 10 when x 5 2. ^8.y 5 6 when x 5 18.

9. y 5 2 when x 5 5. ^10.y 5 9.92 when x 5 12.8.

11. y 5 1.85 when x 5 0.925. ^12.y 5 1²₉ when x5 3²₃.

Graph each direct variation equation.

13. y 5 5x ^{14. y} 5 2²₅x 15. y 5³₄x

16. An equilateral triangle is a triangle with three equal sides. Th e perimeter of an equilateral triangle varies directly with the length of one side. What is an equation that relates the perimeter p and length l of a side? What is the graph of the equation?

17. Th e amount a you fi ll a tub varies directly with the amount of time t you fi ll it. Suppose you fi ll 25 gallons in 5 minutes. What is an equation that relates a and t? What is the graph of

the equation?

no

y 5 5x; 40

y 5 0.775x; 6.2 y 5 2x; 16

y 5²₅x; ¹⁶₅

y 5¹₃x; ⁸₃

y 5¹₃x; ⁸₃ yes; 2¹₃

yes; 2¹₄

yes; 2²₅ yes; 2¹₃ no

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

4 8 12

2 4 6

O

E p

Side length

Perimeter

p 5 3l

a 5 5t

10 20 30

2 4 6

O

Time (min)

Amount (sal)

(14)

5-2 ^Practice

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.

18. 19. 20.

Suppose y varies directly with x. Write and graph a direct variation equation that relates x and y.

21. y5 26 when x 5 3. ^22.y5 2⁴₃ when x5 24. ^23.y5 ⁵₈ when x5 ¹₂.

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

24. the total number of miles run and the number of miles you run per day when training for a race

25. Jackson’s age and Dylan’s age

26. a recipe that calls for 2 cups of sugar for each cup of fl our

27. Writing In a direct variation equation, describe how the slope of the graph of the line is related to the constant of variation.

28. Janine gets paid $16.75 per hour at her job. Write a direct variation equation where h represents the number of hours she works and d represents the amount of money she earns. Graph the equation.

x y

2 27 5

22.5 26.25

8.75

x y

9 12 23

10.8 3.6 14.4

x y

25.2 26.5 4.8

219.5 215.6 14.4

yes; y 5 21.25x no yes; y 5 3x

y 5 22x

yes; the total will be the number of days times the miles run per day.

no; the difference in their age is constant, but the ratio is not.

yes; for every cup of ﬂ our, use 2 cups of sugar.

They are equal.

y 5⁵₄x y 5¹₃x

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

d 5 16.75 h

2 100 200 300

4 6 8 10 12 14 O

h Hours

Money earned ($)

(15)

5-2 ^Practice Direct Variation

^{Form K} Determine whether each equation represents a direct variation. If it does, fi nd

the constant of variation.

1. 3y1 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 x5 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 5²₃ 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?

p 5 4l

no yes; ²₅

yes; ¹

8 yes; 2

y 5¹₂x; 4 y 5 3x; 24

y 5³₈x; 3 y 5 3.6x; 28.8

Ź4 Ź2 2 4

Ź2 Ź4 2 4

x y

O Ź4 Ź2 2 4

Ź2 Ź4 2 4

x y

O Ź4 Ź2 2 4

Ź2 Ź4 2 4

x y

O

Ź8 Ź4 4 8

Ź4 Ź8 4 8

x y

O

(16)

5-2 ^Practice

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 5¹⁵₂ 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 Ľ2

5.4 2.7 Ľ2.7

x y

6 10 Ľ7

Ľ6.9 Ľ11.5 Ľ8.05

x y

no; Sara’s hourly wages are Pasco’s hourly wages plus 2

yes; To ﬁ nd the total number of meals provided, you multiply the number of days by 3.

Yes; To ﬁ nd the total number of points Jasmine has scored, multiply the number of games played by 10.

If the line passes through the origin, then it is a direct variation.

yes; y 5 1.35x no

y 5 23x y 5 2³₂x

(17)

5-2 Standardized Test Prep

Direct 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.

x O

4 y

2 2

Ź4 Ź2

Ź4 Ź2 4

21

201.6

200

1 4 3

4

98 675 43 10

1 2

98 675 4 21 0

98 675 43210

98 7 453210

98 675 4321

98 76 54 32 10 2

3 6

0 2

98 76 453 10

. 6 1 0 2

98 76 45 21 0

98 76 453210

98 7 453210

98 76 45321

98 675 43210 2

3 6

0 2

98 675 43 10

4 / 3

98 675 4 21 0

98 675 43210

98 7 453210

98 675 4321

98 76 54 32 10 2

3 6

0 2

98 675 43 10

98 675 4 21 0

98 675 43210

98 7 453210

98 675 4321

98 76 54 32 10 2

3 6

0 2

4 / 1

98 76 453 10

0 0 2

98 76 45 21 0

98 76 453210

98 7 453210

98 76 54 32 1

98 675 43210 2

3 6

0 2

(18)

5-2 ^Enrichment Direct Variation

A rubber ball is dropped out a window. Th e height that the ball bounces varies directly with the height of that window. Th is relation is modeled by the equation y 5 0.4x, in which x represents the height from which the ball is dropped in meters and y represents the height the ball bounces in meters.

1. Th e ball is dropped from a height of 25 m above the ground. How high does it bounce?

2. Th e ball bounces 13.5 cm in the air. From what height was it dropped?

3. List at least three factors other than starting height that could aff ect the fi nal height of the ball.

4. How might each of these factors change the equation? Would the new

equation be a direct variation? If so, come up with a new constant of variation that makes sense given the factor(s) you have identifi ed, and answer

Exercises 1 and 2 for your new direct variation.

5. Draw the graph of the original ball. Graph at least one of your new functions on the same set of axes.

6. What can you say regarding the largest possible value for the constant of variation given the situation described? Why does this limit exist? (What would happen if the limit didn’t exist?)

7. Th e original rubber ball is dropped from the same distance, but hits the roof of a building 20 ft off the ground. Is the resulting function still a direct variation?

Explain.

10 m

33.75 cm

Answers may vary. Sample: type of ball; whether you drop the ball or throw the ball down; wind resistance

Answers may vary. Sample: Suppose a ball bounced to 70% of its starting height.

y 5 0.7x, 17. 5 m, 19.3 cm

Answers may vary. Sample:

The maximum constant for variation is 1; if it were more than 1, the ball would have bounced higher.

It will rebound y 5 0.4(x 2 20) feet above that roof, or y 5 0.4(x 2 20) 1 20 feet above the ground. This equation, y 5 0.4x 1 12, is not direct variation.

20 40 60

20 40 60 O

x y

Start Height

Bounce Height

Original Bouncer ball

(19)

5-2 ^Reteaching Direct Variation

A direct variation is a relationship that can be represented by a function in the form y5 kx where k 2 0. Th e constant of variation for a direct variation k is the coeffi cient of x. Th e equation y 5 kx can also be written as ^y_x 5 k.

Problem

Does the equation 6x 1 3y 5 9 represent a direct variation? If so, fi nd the constant of variation.

If the equation represents a direct variation, the equation can be rewritten in the form y5 kx. So, solve the equation for y to determine whether the equation can be written in this form.

6x 1 3y 5 9

3y5 9 2 6x Subtract 6x from each side.

y 5 3 2 2x Divide each side by 3.

You cannot write the equation in the form y5 kx. So 6x 1 3y 5 9 does not represent a direct variation.

Problem

Does the equation 5y 5 3x represent a direct variation? If so, fi nd the constant of variation.

Again, if the equation represents a direct variation, the equation can be rewritten in the form y5 kx. So, solve the equation for y to determine whether the equation can be written in this form.

5y5 3x

y5 ³₅ x Divide each side by 5.

Th e equation has the form y 5 kx, so the equation represents a direct variation.

Th e coeffi cient of x is ³₅, so the constant of variation is ³₅.

Exercises

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

1. 2y5 x ^2.3x1 2y 5 1 ^3.24y 5 8x

4. 2x5 y 2 5 ^5.4x2 3y 5 0 ^6.5x5 2y

yes; ¹₂

yes; ⁴₃ yes; ⁵₂

no

yes; 22

(20)

5-2 Reteaching

(continued)

Direct Variation

To write an equation for direct variation, fi nd the constant of variation k using an ordered pair. Th en use the value of k to write an equation.

Problem

Suppose y varies directly with x, and y5 24 when x 5 8. What direct variation equation relates x and y? What is the value of y when x5 10?

You are given that x and y vary directly. Th is means that the relationship between x and y can be written in the form y 5 kx, where k is a constant.

y 5 kx Start with the direct variation equation.

245 k(8) Substitute the given values: 8 for x and 24 for y.

35 k Divide each side by 8 to solve for k.

y 5 3x Write the direct variation equation that relates x and y by substituting 3 for k in y 5 kx.

Th e equation y 5 3x relates x and y. When x 5 10, y 5 3(10)or 30.

Exercises

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 6.

7. y5 14 when x 5 2. ^8.y5 3 when x 5 9.

9. y5 12 whenx 5 224. ^10.y5 281 when x 5 9.

11. y5 216 when x 5 24. ^12.y5 5 when x 5 20.

13. Consider the direct variation y 5 3x.

a. List three ordered pairs that satisfy the equation.

b. Plot your three ordered pairs from part (a) on a coordinate grid.

c. Complete the graph of y5 3x on the grid.

y 5 7x; 42

y 5 29x; 254 y 5 4x; 24

Answers may vary. Sample: (0,0), (1, 3), (2. 6)

Sample:

y 5¹₃x; 2 y 5 2¹₂x; 23

y 5¹₄x; ³₂

2 4 6

O

x y

(0, 0) (1, 3)

(2, 6)

(21)

Complete the vocabulary chart by filling in the missing information.

Word or

Word Phrase Definition Picture or Example

linear equation An equation that models a linear function

y 5 2x

linear parent

function y 5 x or f (x) 5 x

parent function y 5 x of

y 5 x, y 5 2x, and y 5 3x

slope-intercept

form An equation of the form y 5 mx 1 b, where m is the slope and b is the y-intercept

y-intercept y 5 2x 1 1

5-3 Additional Vocabulary Support

Slope - Intercept Form

1.

2.

3.

4.

y-intercept A family of functions is a

group of functions with common characteristics.

A parent function is the simplest function with these characteristics.

y-coordinate of a point where the graph crosses the y-axis

y 5 mx 1 b y 5 3x 1 4

y

x 24 22 O

22 2 4

24

2 4

(22)

5-3 Think About a Plan

Slope-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?

Understanding the Problem

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

Planning the Solution

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?

yes; the rate of change (10 pieces/min) is constant

175; the y-intercept

number of pieces placed and change in time; 10

y5 175 1 10x

675 pieces

200 400 600

20 40 Time (min)60 O

x y

(23)

5-3 ^Practice Slope-Intercept Form

^{Form G} Find the slope and y-intercept of the graph of each equation.

1. y 5 3x 2 5 ^2.y 5 25x 1 13 ^3.y 5 2x 2 1

4. y 5 211x 1 6 ^5.y 5 25 ^6.y 5¹₂x 1 6

7. y 5 26.75x 1 8.54 ^8.y 5 2²₃x 2¹₉ ^9.y 5 2.25

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

10. m 5 21, b 5 3 ^11.m 5 4, b 5 22 ^12.m 5 25, b 5 28 13. m 5 0.25, b 5 6 ^14.m 5 0, b 5 211 ^15.m 5 1, b 5³₈

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

16. 17. 18.

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

19. (3, 5) and (0, 4) 20. (2, 6) and (24, 22) 21. (21, 3) and (23, 1)

22. (27, 5) and (3, 0) ^23.(10, 2) and (22, 22) ^24.(0, 21) and (5, 6)

25. (3, 2) and (21, 6) ^26.(24, 23) and (3, 4) ^27.(2, 8) and (23, 6) 6 y

4 2

Ź2

x

O 2

Ź4 Ź2 4

x O

4 y

2 2

Ź4 Ź2

Ź4 Ź2 4

x O

2 y

2

Ź6 Ź4 Ź2

Ź4 Ź2 4

3; 25

211; 6

26.75; 8.54 0; 2.25

y 5 2x 1 3

y 5 0.25x 1 6 y 5 211

y 5 2x 1 1 y 5 25 y 5 2¹₂x 1 4

y 5 x 1³₈ y 5 25x 2 8 y 5 4x 2 2

2²₃; 2¹₉ 25; 13 0; 25

21; 21

1 2; 6

y 5¹₃x 1 4 y 5⁴₃x 1¹⁰₃

y 5²₅x 1³⁶₅ y 5 2¹₂x 1³₂

y 5 2x 1 5 y 5 x 1 1

y 5¹₃x 2⁴₃ y 5⁷₅x 2 1 y 5 x 1 4

(24)

5-3 ^Practice

Slope-Intercept Form

Graph each equation.

28. y5 x 1 3 ^29.y5 4x 2 1 ^30.y5 2x 1 6

31. y5 3x 2 2 ^32.y5 25x 1 1 ^33.y5 27x 2 4

34. Hudson is already 40 miles away from home on his drive back to college. He is driving 65 mi/h. Write an equation that models the total distance d travelled after h hours. What is the graph of the equation?

35. When Phil started his new job, he owed the company $65 for his uniforms. He is earning $13 per hour. Th e cost of his uniforms is withheld from his earnings. Write an equation that models the total money he has m after h hours of work. What is the graph of the equation?

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

36. y1 4 5 26x ^37.y1 ¹₂ x 5 24 ^38.3y2 12x 1 6 5 0

39. y2 5 5 ¹₃(x 2 9) ^40.y2 ²₅ x 5 0 ^41.2y1 6a 2 4x 5 0

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

x y

4 8 Ź4

Ź8 8

Ź4 Ź8 4 O

x y

4 8 Ź4

Ź8 8

Ź4 Ź8 4 O

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

x y

2 4 Ź2

Ź4 4

Ź2 Ź4 2 O

m 5 26; b 5 24 m 5 4; b 5 22

m 5 2; b 5 23a m 5 2¹₂; b 5 24

m 5¹₃; b 5 2

d 5 65h 1 40

m 5 13h 2 65

m 5²₅; b 5 0

200 400 600

2 4 6

O

h d

n m

4 8 Ź4

Ź8 80

Ź40 Ź80 40 O

(25)

5-3 ^Practice Slope-Intercept Form

^{Form K} Find the slope and y-intercept of the graph of each equation.

1. y5 22x 1 7 ^2.y5 6x 1 11

3. y5 27x 2 8 ^4.y5 22.5x 1 3.2

5. y5 29 ^6.y5 ¹₄ x 2²₇

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

7. m5 25, b 5 26 ^8.m5 1, b 5 24

9. m5 0.4, b 5 29 ^10.m5 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

O Ź2

Ź2 2 2

x y

O Ź2

Ź2 2 2

m 5 22; b 5 7 m 5 6; b 5 11

m 5 27; b 5 28 m 5 22.57; b 5 3.2

m 5 0; b 5 29 m 5¹₄; b 5 2²₇

y 5 25x 2 6 y 5 x 2 4

y 5 0.4x 2 9 y 5 3

y 5 2x 1 1 y 5 21

y 5 22x y 5 2x 1 7

y 5 2¹₂x 1 16 y 5 22x 2 9

(26)

5-3 ^Practice

Slope-Intercept Form

Graph each equation.

17. y5 x 2 2 ^18.y5 3x 1 1

19. y5 2x 2 1 ^20.y5 23x 2 2

21. y5 ¹₂ x 1 2 ^22.y5 2⁴₅ x2 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?

d 5 45h

Ź4 Ź2 2 4

Ź2 Ź4 2 4

x y

O Ź4 Ź2 2 4

Ź2 Ź4 2 4

x y

O

Ź4 Ź2 2 4

Ź2 Ź4 2 4

x y

O Ź4 Ź2 2 4

Ź2 Ź4 2 4

x y

O

Ź4 Ź2 2 4

Ź2 Ź4 2 4

x y

O Ź8 Ź4 4 8

Ź4 Ź8 4 8

x y

O

1 2 3 4 5

100 150

50 200 250

x y

O

(27)

5-3 Standardized Test Prep

Slope-Intercept Form

Multiple Choice

For Exercises 1–5, choose the correct letter.

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

A. y 5 2³₂ x 1 4 ^C.y5 2²₃x1 4

B. y 5²₃ x 1 4 ^D.y5 2²₃x1 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.y5 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.y5 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 5¹₂ x 1 12 ^H.y5 2x 1 9

G. y 5¹₂ x 1 6 ^I.y5 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 2³₄? What is the graph of the equation?

6 y 4 2

Ź2

x

O 2

Ź4 Ź2 4

x y

2 6 10

13 17 15 C

G

D

F

B

y 5 2³₄x 2 4

[2] Both parts answered correctly.

[1] One part answered correctly.

[0] Neither part answered correctly.

x y

4 8 Ź4

Ź8 8

Ź4 Ź8 4 O

Name Class Date. Additional Vocabulary Support

5-1 Additional Vocabulary Support

Rate of Change and Slope

5-1 Think About a Plan

Rate of Change and Slope

5-1 Practice Rate of Change and Slope

5-1 Practice

Rate of Change and Slope

5-1 Practice Rate of Change and Slope

5-1 Practice

Rate of Change and Slope

5-1 Standardized Test Prep

Rate of Change and Slope

Short Response

5-1 Enrichment Rate of Change and Slope

5-1 Reteaching Rate of Change and Slope

5-1 Reteaching

Rate of Change and Slope

Exercises

Exercises

5-2 Additional Vocabulary Support

Direct Variation

5-2 Think About a Plan

Direct Variation

Understanding the Problem

Planning the Solution

Getting an Answer

5-2 Practice Direct Variation

5-2 Practice

Direct Variation

5-2 Practice Direct Variation

5-2 Practice

Direct Variation

5-2 Standardized Test Prep

Direct Variation

Gridded Response

5-2 Enrichment Direct Variation

5-2 Reteaching Direct Variation

Exercises

5-2 Reteaching

Direct Variation

Exercises

5-3 Additional Vocabulary Support

Slope - Intercept Form

5-3 Think About a Plan

Slope-Intercept Form

Understanding the Problem

Planning the Solution

Getting an Answer

5-3 Practice Slope-Intercept Form

5-3 Practice

Slope-Intercept Form

5-3 Practice Slope-Intercept Form

5-3 Practice

Slope-Intercept Form

5-3 Standardized Test Prep

Slope-Intercept Form

Multiple Choice

Short Response

5-1 ^Practice Rate of Change and Slope

5-1 ^Practice

5-1 ^Practice Rate of Change and Slope

5-1 ^Practice

5-1 ^Enrichment Rate of Change and Slope

5-1 ^Reteaching Rate of Change and Slope

5-2 ^Practice Direct Variation

5-2 ^Practice

5-2 ^Practice Direct Variation

5-2 ^Practice

5-2 ^Enrichment Direct Variation

5-2 ^Reteaching Direct Variation

5-3 ^Practice Slope-Intercept Form

5-3 ^Practice

5-3 ^Practice Slope-Intercept Form

5-3 ^Practice