1. Lesson 5-2 Slope and Direct Variation

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4. Objectives Write and graph direct variation equations Solve problems involving direct variation

5. Vocabulary Constant of variation – the number k in equations the form y = kx Graph – is a plot of an ordered pair (x,y) on the coordinate plane Family of graphs – graphs and equations of graphs that have at least one characteristic in common Parent graph – the simplest of the graphs in a family of graphs

6. y x y x x-y Coordinate Plane Slope, m (∆y / ∆x)Point Plotting (x, y) (-4, 7) (5, -8) x – left or right y – up or down right 5 down 8 left 4 up 7 (x, y) (-4, 7) (5, -8) rise: ∆y – change in y (7 - -8 = 15) run: ∆x – change in x (-4 – 5 = -9) ∆y = 15 ∆x = -9 5 m = ------ -3 rise run

7. Example 1a Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points. Slope formula Simplify. Answer: The constant of variation is 2. The slope is 2.

8. Example 1b Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points. Slope formula Simplify. Answer: The constant of variation is –4. The slope is –4.

9. Example 2 Step 1Write the slope as a ratio. Step 2Graph (0, 0). Step 3From the point (0, 0), move up 1 unit and right 1 unit. Draw a dot. Step 4Draw a line containing the points.

10. Example 3 Step 1Write the slope as a ratio. Step 2Graph (0, 0). Step 3From the point (0, 0), move down 3 units and right 2 units. Draw a dot. Step 4Draw a line containing the points.

11. Example 4 Suppose y varies directly as x, and y = 9when x = -3 Write a direct variation equation that relates x and y. Find the value of k. Direct variation formula Replace y with 9 and x with –3. Divide each side by –3. Simplify. Answer:Therefore,

12. Example 5 Travel The Ramirez family is driving cross-country on vacation. They drive 330 miles in 5.5 hours. Write a direct variation equation to find the distance driven for any number of hours. WordsThe distance traveled is 330 miles, and the time is 5.5 hours. Variables Distance equals rate times time. Equation 330 mi r5.5h

13. Example 5 cont Solve for the rate. Answer:Therefore, the direct variation equation is Original equation Divide each side by 5.5. Simplify.

14. Summary & Homework Summary: –A direct variation is described by an equation of the form y = kx, where k ≠ 0 –In y = kx, k is the constant of variation. It is also the slope of the related graph Homework: –Pg 268 16-46 even