12-5 Direct Variation Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Warm Up Use 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 Direct Variation (9, 3), – 1 7 (5, –2), 2 3 (–7, 5), – 1 4

Problem of the Day Where do the lines defined by the equations y = –5x + 20 and y = 5x – 20 intersect? (4, 0) Course Direct Variation

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

Vocabulary direct variation constant of proportionality Insert Lesson Title Here Course Direct Variation

Course Direct Variation

Course 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 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 Direct Variation

You can also compare ratios to see if a direct variation occurs = ? ≠ 264 The ratios are not proportional. The relationship of the data is not a direct variation. Additional Example 1A Continued Course Direct Variation

Determine whether the data set shows direct variation. Additional Example 1B: Determining Whether a Data Set Varies Directly Course 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 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 === Compare ratios. Additional Example 1B Continued Course Direct Variation

Determine whether the data set shows direct variation. Check It Out: Example 1A Kyle's Basketball Shots Distance (ft) Number of Baskets530 Course 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 Number of Baskets Distance (ft) Course Direct Variation

You can also compare ratios to see if a direct variation occurs. Check It Out: Example 1A Continued = ?  60. The ratios are not proportional. The relationship of the data is not a direct variation. Course Direct Variation

Determine whether the data set shows direct variation. Check It Out: Example 1B Ounces in a Cup Ounces (oz) Cup (c)1234 Course Direct Variation

Make a graph that shows the relationship between ounces and cups. Check It Out: Example 1B Continued Number of Cups Number of Ounces Course 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 Direct Variation The ratios are proportional. The relationship is a direct variation. Compare ratios. = 1 8 ==

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 = kx 54 = k  6 9 = k y = 9x y varies directly with x. Substitute for x and y. Solve for k. Substitute 9 for k in the original equation. Course Direct Variation

x is 12 when y is 15 Additional Example 2B: Finding Equations of Direct Variation y = kx 15 = k  12 y varies directly with x. Substitute for x and y. Solve for k. = k 5 4 Substitute for k in the original equation. 5 4 y = x 5 4 Course 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  4 6 = 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 Direct Variation

x is 28 when y is 14 Check It Out: Example 2B y = kx 14 = k  28 y varies directly with x. Substitute for x and y. Solve for k. = k 1 2 Substitute for k in the original equation. 1 2 y = x 1 2 Course 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 Direct Variation

Additional Example 3 Continued interest from CD and time interest from CD time = 17 1 = = 17 interest from CD time 34 2 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. = = = 17 interest from CD time = Course Direct Variation

Additional Example 3 Continued interest from money market and time interest from money market time = = interest from money market time = = ≠ 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 Direct Variation

Mr. Ortega has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation. Check It Out: Example 3 Course Direct Variation InterestInterest from Time (mo)from CD ($)Money Market ($)

Check It Out: Example 3 Continued interest from CD time = 12 1 interest from CD time = = 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 Direct Variation

Check It Out: Example 3 Continued B. interest from money market and time interest from money market time = = interest from money market time = = ≠ 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 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 x is 45 when y is y is 6 when x is 5. y = 26x Insert Lesson Title Here y = x Course 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 Direct Variation