2. WARM UP 1. What is the slope of the line through the points (4,3) and (11,5)? a.7/2 b.-2/7 c.2/7 d.-7/2 2. What time is on this clock? 3

3. WARM UP 1. What is the slope of the line through the points (4,3) and (11,5)? a.7/2 b.-2/7 c.2/7 d.-7/2 2. What time is on this clock? 2

4. WARM UP 1. What is the slope of the line through the points (4,3) and (11,5)? a.7/2 b.-2/7 c.2/7 d.-7/2 2. What time is on this clock? 1

5. WARM UP 1. What is the slope of the line through the points (4,3) and (11,5)? a.7/2 b.-2/7 c.2/7 d.-7/2 2. What time is on this clock? 0

6. When two quantities y and x have a constant ratio k, they are said to have direct variation. The constant k is called the constant of variation. Model for Direct Variation: y = kx or = k, where k = 0 4.6 Direct Variation yxyx

7. EXAMPLE 1 Write a Direct Variation Model The variables x and y vary directly. One pair of values is x = 5 and y = 20 a.Write an equation that relates x and y. b.Find the value of y when x =12 SOLUTION a.Because x and y vary directly, the equation is in the form of y = kx STEP ONE: Write model for direct variationy = kx STEP TWO: Substitute 5 for x and 20 for y20 = k(5) STEP THREE: Divide each side by 54 = k

8. 4.6 Direct Variation EXAMPLE 1 Write a Direct Variation Model The variables x and y vary directly. One pair of values is x = 5 and y = 20 a.Write an equation that relates x and y. b.Find the value of y when x =12 SOLUTION b.Find the value of y when x = 12 STEP ONE: Substitute 12 for xy = 4(12) STEP TWO: Simplifyy = 48 ANSWER> When x = 12, y = 48

9. 4.6 Direct Variation CHECK POINT Write a Direct Variation Model The variables x and y vary directly. Use the given values to write a direct variation model that relates x and y. 1.x =2, y = 6 2.x =3, y = 21 3.x =8, y = 96

10. GRAPHING DIRECT VARIATION MODELS Because x = 0 and y = 0 is a solution of y = kx, the graph of a direct variation equation is always a line through the origin. 4.6 Direct Variation Origin (D’uh!)

11. 4.6 Direct Variation EXAMPLE 2 Graph a Direct Variation Model Graph the equation y = 2x SOLUTION Plot a point at the origin. 1

12. 4.6 Direct Variation EXAMPLE 2 Graph a Direct Variation Model Graph the equation y = 2x SOLUTION Find a second point by choosing any Value for x and substituting it into the Equation. (Use the value 1) STEP ONE: Write the original equation y = 2x STEP TWO: Substitute 1 for x y = 2(1) STEP THREE: Simplify. The y-value is 2. ANSWER> The second point is (1,2) 2

13. 4.6 Direct Variation EXAMPLE 2 Graph a Direct Variation Model Graph the equation y = 2x SOLUTION Plot the second point and draw a Line through the origin and the second point. ANSWER> The graph of y = 2x is shown above. 3

14. 4.6 Direct Variation CHECK POINT Graph a Direct Variation Model Graph the equation 1.y = x 2.y = -2x 3.y = 3x

15. 4.6 Direct Variation EXAMPLE 3 Use a Direct Variation Model FORT KNOX The gold stored in Fort Knox is in the form of standard mind bars called bullion, of almost pure gold. Given that 5 gold bars weigh 137.5 pounds, find the weight of 36 gold bars. SOLUTION a.Begin by writing a model that relates the weight W to the number n of gold bars. STEP ONE: Write model for direct variationW = kn STEP TWO: Substitute 137.5 for W and 5 for n137.5 = k(5) STEP THREE: Divide each side by 527.5 = k

16. 4.6 Direct Variation EXAMPLE 3 Use a Direct Variation Model FORT KNOX The gold stored in Fort Knox is in the form of standard mind bars called bullion, of almost pure gold. Given that 5 gold bars weigh 137.5 pounds, find the weight of 36 gold bars. SOLUTION b.A direct variation model for the weight of a gold bar is W = 27.5n. Use the model to find the weight of 36 gold bars. STEP ONE: Substitute 36 for nW = 27.5(36) STEP TWO: SimplifyW = 990 ANSWER> Thirty six gold bars weigh 990 pounds. YOU’RE CERTIFIED

17. 4.5 The Slope of a Line CLASSWORK/HOMEWORK Page 239 #s 3 -24