November 30, direct variation ink.notebook. page 162. page Direct Variation. page 163. page 164 page 165

Transcription

1 4.6 direct variation ink.notebook page 161 page Direct Variation page 163 page 164 page 165 1

2 Lesson Objectives Standards Lesson Notes Lesson Objectives Standards Lesson Notes 4.6 Direct Variation F.IF.1 F.IF.7 I will determine if a function is a direct variation function I will graph a direct variation function Press the tabs to view details. Press the tabs to view details. Lesson Objectives Standards Lesson Notes A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Direct Variation y = kx where k = 0 is read: y varies directly as x if x increases then y increases if x decreases then y decreases k = constant of variation Press the tabs to view details. k = y x 2

3 4.6 direct variation ink.notebook a graph that shows direct variation always goes through the origin Direct Variation What is it? A linear equation that passes through the origin (0, 0) Constant? y k= x k is the constant of variation is constant for ALL ordered pairs Equation? y = kx y =1x 2 yes, a direct variation no, not a direct variation where k = 0 Graph: y Table: y x x Example: The weekly salary a woman earns, S, varies directly as the number of hours, h, which she works. S = kh k = her hourly salary If she makes $7 per hour: S = 7h 3

4 4.6 direct variation ink.notebook Example B: 1) Substitute y and x in If y = 9 when x = 3, find y when x = 5. 2) Divide and Solve for k y = kx - 9 = k(3) 9 = k(3) Graph each line and then determine if it has direct variation. 2. y = 3x y = - 3x y = -3(-5) y = 15 3) Write general equation y = x k = -3 4) Substitute in last given value Suppose y varies directly as x. Write a direct variation equation that relates x to y. Then solve. 3. If y = 4 when x = 2, find y when x = 16. y4 = k x2 4

5 4. If y = 9 when x = 3, find x when y = If y = 4.8 when x = 1.6, find x when y = 24. y = k x The total cost C of gasoline is $3.00 times the number of gallons g. Write a direct variation equation that relates the variables. 5

6 4.6 direct variation ink.notebook The distance formula d = rt is a direct variation equation. In the formula, distance d varies directly as time t, and the rate r is the constant of variation. 7. A family drove their car 225 miles in 5 hours. a) Write a direct variation equation to find the distance traveled for any number of hours b) Estimate how many hours it would take the family to drive 360 miles. 8. Charles's Law states that, at a constant pressure, volume of a gas V varies directly as its temperature T. A volume of 4 cubic feet of a certain gas has a temperature of 200 degrees Kelvin. a) Write a direct variation equation that relates the variables. b) Find the volume of the same gas at 250 degrees Kelvin. 6

7 On Your Whiteboards Graph each line and then determine if it has direct variation. a) y = x + 0 b) c) If y = 8 when x = 2, find y when x = 8. Suppose y varies directly as x, and y = 30 when x = 5. Write a direct variation equation that relates x and y. Use the direct variation equation to find x when y = 18. y 30 k 5 6 = x y = k x 18 = k 6 x y = k x 8 2 k = x = 7

8 d) The total cost C of bulk jelly beans is $4.49 times the number of pounds p. a) Write a direct variation equation that relates the variables. b) Find the cost of pound of jelly beans. On the Worksheet Do the following graphs show direct variation? Write for each Homework 8

9 Do the following graphs show direct variation? Write for each. 3. Graph each line and then determine if it has direct variation. 4. y = 2x Graph each line and then determine if it has direct variation. 6. y = x Suppose y varies directly as x. Write a direct variation equation that relates x to y. Then solve. 7. If y = 45 when x = 15, find x when y = 15. 9

10 8. If y = 4 when x = 2, find y when x = If y = 9 when x = 3, find y when x = If y = 4 when x = 16, find y when x = If y = 72 when x = 8, find x when y =

11 12. If y = 2 when x = 4, find y when x = 10. Write a direct variation equation that relates the variables. 13. The weight W of an object is 9.8m/s2 times the mass of the object m. 14. Music downloads are $0.99 per song. The total cost of d songs is T. 15. The circumference of a circle C is approximately 3.14 times the diameter d. 16. The distance a jet travels varies directly as the number of hours it flies. A jet traveled 3420 miles in 6 hours. a) Find the constant of variation, k. 17. The total cost of tickets to a local concert varies directly as the number of tickets you buy. 4 tickets cost $72. a) Find the constant of variation, k. b) Write a direct variation equation for the distance d flown in time t. b) Write an equation to show this direct variation c) Estimate how many hours it will take for an airliner to fly 6500 miles. Round to nearest tenth. c) Find the cost of 11 tickets. 11

12 Find the slope of each pair of points. 18. (7, 1) (5, 2) 19. (4, 2) ( 3, 2) Answers: 1) yes 3) no 5) no 7) k = 3, y = 3x, x = 5 9) k = 3, y = 3x, y = 15 11) k = 9, y = 9x, x = 7 13) w = 9.8m 15) C = 3.14d 17) a) k = $18 per ticket b) y = 18x c) $198 19) 0 12