12-5 Direct Variation

Course 3

Warm UpWarm UpProblem of the DayProblem of the DayLesson PresentationLesson Presentation

Warm UpUse the point-slope form of each equation to identify a point the line passes through and the slope of the line.1. y – 3 = – (x – 9)2. y + 2 = (x – 5)3. y – 9 = –2(x + 4)4. y – 5 = – (x + 7)

(–4, 9), –2

Course 3

12-5 Direct Variation

17

23

14

(9, 3), – 17

(5, –2), 23

(–7, 5), – 14

Problem of the DayWhere do the lines defined by the equations y = –5x + 20 and y = 5x – 20 intersect?(4, 0)

Course 3

12-5 Direct Variation

Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

Course 3

12-5 Direct Variation

Vocabularydirect variationconstant of proportionality

Insert Lesson Title Here

Course 3

12-5 Direct Variation

Course 3

12-5 Direct Variation

Course 3

12-5 Direct Variation

The graph of a direct-variation equation is always linear and always contains the point (0, 0). The variables x and y either increase together or decrease together.

Helpful Hint

Determine whether the data set shows direct variation.

Additional Example 1A: Determining Whether a Data Set Varies Directly

Course 3

12-5 Direct Variation

Make a graph that shows the relationship between Adam’s age and his length. The graph is not linear.

Additional Example 1A Continued

Course 3

12-5 Direct Variation

You can also compare ratios to see if a direct variation occurs.

223

2712=?

81

264

81 ≠ 264

The ratios are not proportional.

The relationship of the data is not a direct variation.

Additional Example 1A Continued

Course 3

12-5 Direct Variation

Determine whether the data set shows direct variation.

Additional Example 1B: Determining Whether a Data Set Varies Directly

Course 3

12-5 Direct Variation

Make a graph that shows the relationship between the number of minutes and the distance the train travels.

Additional Example 1B Continued

Plot the points.The points lie in a straight line.

Course 3

12-5 Direct Variation

(0, 0) is included.

You can also compare ratios to see if a direct variation occurs.

The ratios are proportional. The relationship is a direct variation.

2510

5020

7530

10040= = = Compare ratios.

Additional Example 1B Continued

Course 3

12-5 Direct Variation

Determine whether the data set shows direct variation.

Check It Out: Example 1A

Kyle's Basketball Shots Distance (ft) 20 30 40Number of Baskets 5 3 0

Course 3

12-5 Direct Variation

Make a graph that shows the relationship between number of baskets and distance. The graph is not linear.

Check It Out: Example 1A Continued

Num

ber o

f Bas

kets

Distance (ft)

2

3

4

20 30 40

1

5

Course 3

12-5 Direct Variation

You can also compare ratios to see if a direct variation occurs.

Check It Out: Example 1A Continued

520

330=?

60

150

150 60.

The ratios are not proportional.

The relationship of the data is not a direct variation.

Course 3

12-5 Direct Variation

Determine whether the data set shows direct variation.

Check It Out: Example 1B

Ounces in a CupOunces (oz) 8 16 24 32Cup (c) 1 2 3 4

Course 3

12-5 Direct Variation

Make a graph that shows the relationship between ounces and cups.

Check It Out: Example 1B Continued

Num

ber o

f Cup

s

Number of Ounces

2

3

4

8 16 24

1

32

Course 3

12-5 Direct Variation

Plot the points.The points lie in a straight line.

(0, 0) is included.

You can also compare ratios to see if a direct variation occurs.

Check It Out: Example 1B Continued

Course 3

12-5 Direct Variation

The ratios are proportional. The relationship is a direct variation.

Compare ratios. = 1 8 = =2

163

24 432

Find each equation of direct variation, given that y varies directly with x.y is 54 when x is 6

Additional Example 2A: Finding Equations of Direct Variation

y = kx54 = k 6

9 = ky = 9x

y varies directly with x.Substitute for x and y.Solve for k.Substitute 9 for k in the original equation.

Course 3

12-5 Direct Variation

x is 12 when y is 15

Additional Example 2B: Finding Equations of Direct Variation

y = kx15 = k 12

y varies directly with x.Substitute for x and y.

Solve for k. = k54

Substitute for k in the original equation.

54y = x5

4

Course 3

12-5 Direct Variation

Find each equation of direct variation, given that y varies directly with x.y is 24 when x is 4

Check It Out: Example 2A

y = kx

24 = k 46 = k

y = 6x

y varies directly with x.

Substitute for x and y.

Solve for k.

Substitute 6 for k in the original equation.

Course 3

12-5 Direct Variation

x is 28 when y is 14

Check It Out: Example 2B

y = kx

14 = k 28y varies directly with x.

Substitute for x and y.

Solve for k. = k12

Substitute for k in the original equation.

12y = x1

2

Course 3

12-5 Direct Variation

Mrs. Perez has $4000 in a CD and $4000 in a money market account. The amount of interest she has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.

Additional Example 3: Money Application

Course 3

12-5 Direct Variation

Additional Example 3 Continuedinterest from CD and time

interest from CDtime = 17

1 = = 17interest from CDtime

342

The second and third pairs of data result in a common ratio. In fact, all of the nonzero interest from CD to time ratios are equivalent to 17.

The variables are related by a constant ratio of 17 to 1, and (0, 0) is included. The equation of direct variation is y = 17x, where x is the time, y is the interest from the CD, and 17 is the constant of proportionality.

= = = 17interest from CDtime = = 17

1342

513

684

Course 3

12-5 Direct Variation

Additional Example 3 Continued

interest from money market and time

interest from money markettime = = 19

191

interest from money markettime = =18.5

372

19 ≠ 18.5

If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

Course 3

12-5 Direct Variation

Mr. Ortega has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.

Check It Out: Example 3

Course 3

12-5 Direct Variation

  Interest Interest fromTime (mo) from CD ($) Money Market ($)

0 0 01 12 152 30 403 40 454 50 50

Check It Out: Example 3 Continued

interest from CDtime = 12

1interest from CD

time = = 15302

The second and third pairs of data do not result in a common ratio.

If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

A. interest from CD and time

Course 3

12-5 Direct Variation

Check It Out: Example 3 Continued

B. interest from money market and time

interest from money markettime = = 1515

1interest from money market

time = =20 402

15 ≠ 20

If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

Course 3

12-5 Direct Variation

Lesson Quiz: Part I

Find each equation of direct variation, given that y varies directly with x.

1. y is 78 when x is 3.

2. x is 45 when y is 5.

3. y is 6 when x is 5.

y = 26x

Insert Lesson Title Here

y = x19

y = x65

Course 3

12-5 Direct Variation

Lesson Quiz: Part II

4. The table shows the amount of money Bob makes for different amounts of time he works. Determine whether there is a direct variation between the two sets of data. If so, find the equation of direct variation.

Insert Lesson Title Here

direct variation; y = 12x

Course 3

12-5 Direct Variation