1. Direct Variation Honors Math – Grade 8

2. Get Ready for the Lesson It costs $2.25 per ringtone that you download on your cell phone. If you graph the ordered pairs the slope of the line is 2.25 The total cost y depends directly on the number of ringtones that you download x. The rate of change is constant. A direct variation is described by an equation of the form y = kx, where k does not equal 0. The equation y = kx represents a constant rate of change and k is the constant of variation.

3. Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. 1. Find the constant of variation. The equation of the line is y = 3x. The constant of variation, k, is the coefficient of x when the equation is in the form y = kx. The constant of variation is 3. 2. Find the slope of the line. Choose two points on the line and substitute them in the slope formula. Notice that the constant of variation and the slope of the line are the same!

4. Graph each equation. Step 1. Find the slope of the line and write it as a ratio. Remember the slope of the line is the coefficient of x. Step 2. Graph (0, 0). All equations in the form y = kx pass through the origin (0, 0). Step 3. Use the slope to find another point on the line. Move up 4 units (for the rise) and then right 1 unit (for the run). Plot the point. Step 4. Connect the points with a line.

5. Graph each equation. Step 1. Find the slope of the line and write it as a ratio. Remember the slope of the line is the coefficient of x. Step 2. Graph (0, 0). All equations in the form y = kx pass through the origin (0, 0). Step 3. Use the slope to find another point on the line. Move down 1 unit (for the rise) and then right 3 units (for the run). Plot the point. Step 4. Connect the points with a line.

6. KEY CONCEPT Direct Variation Graphs Direct variation equations are in the form:,where and The graph of y = kx always passes through the origin. The slope is positive if k > 0. The slope is negative if k <0

7. Explain whether each relation represents a direct variation. If so, state the constant of variation. -4 -3 -2 1 8 6 4 Find the constant of variation for each ordered pair. Remember… For every ordered pair, k = 2.y varies directly as x. 4 6 10 16 6 9 15 24 Find the constant of variation for each ordered pair. Remember… For every ordered pair, k = 3/2.y varies directly as x.

8. Explain whether each relation represents a direct variation. If so, state the constant of variation. Since a direct variation equation is in the form y = kx, solve the equation for y. Divide both sides of the equation by 3. This equation is not a direct variation equation because it cannot be written in the form y = kx. Since a direct variation equation is in the form y = kx, solve the equation for y. Add x to both sides of the equation. This equation is a direct variation equation because it can be written in the form y = kx. The constant of variation is 1.

9. Suppose y varies directly as x and y = 28 when x = 7. 1. Write a direct variation equation that relates x and y. Use the direct variation formula to find the value of k. Replace x and y with the given values. Solve the equation. Therefore, the direct variation equation is y = 4x. 2. Use the direct variation equation to find x when y = 52. Replace y with 52. Solve the equation. Divide both sides by 4. Therefore, x = 13, when y = 52.

10. Suppose y varies directly as x and y = 6 when x = -18. 1. Write a direct variation equation that relates x and y. Use the direct variation formula to find the value of k. Replace x and y with the given values. Solve the equation. Therefore, the direct variation equation is y = (-1/3)x. 2. Use the direct variation equation to find y when x = -2. Replace x with -2. Solve the equation. Multiply. Therefore, y = -2/3, when x = -2.

11. The migration of snow geese varies directly as the number of hours. A flock of snow geese migrated 375 miles in 7.5 hours. One of the most common application of direct variation is the formula d = rt. Distance (d) varies directly as time (t) and the rate (r) is the constant of variation. 1. Write a direct variation equation. Find the constant of variation. Replace d and t with the given values. The direct variation equation is d = 50t. 2. Graph the equation. 3. Estimate how many hours of flying time it would take the geese to migrate 3000 miles.