Lesson 64

1. Lesson 64 Identifying, writing, and graphing inverse variation

2. Review direct variation • y = kx • As one variableincreases, the other increases at a constant rate, k

3. Inverse variation • Inverse variation is a relationship between 2 variables whose product is a constant. • The equation xy = k or y = k/x, where k is a nonzero constant, defines an inverse variation between x and y • As x increases, y decreases and as x decreases, y increases • Look for the words "varies inversely" or "is inversely proportional to"

4. Example of inverse variation • The relationship between the length and width of a rectangle with a constant area is an inverse variation because in order for the area to stay the same, if the length increases, the width must decrease. • L x W = Area

5. exploration • 1) draw a rectangle that has a width of 1 unit and a length of 16 units on graph paper • 2)draw different rectangles with the same area but different widths and lengths • 3) make a table and complete it after drawing 5 other rectangles with the same area • 4) what happens to the length of each rectangle as the width increases? • 5) what will the product of the width and length always be? • 6)write an equation solved for y showing this relationship.

6. In a direct variation, y is equal to the product of a constant k and x or y = kx • In an inverse variation, y is equal to the quotient of k and x or • y = k/x

7. Identifying an inverse variation • Tell whether each relationship is an inverse variation: • y/6 = x solve for y • 6(y/6) = (x)6 • y = 6x this a direct variation • xy = 5 • y = 5/x inverse variation

8. Direct or inverse variation? • x = 36/y • y/12 = x • 4y=x • 3xy=9

9. Product rule for inverse variation • If (x1, y1) and (x2,y2) are solutions of an inverse variation, then x1y1= x2y2

10. Using the product rule • If y varies inversely as x and y = 3 when x = 12, find x when y = 9 • Use the product rule x1y1= x2y2 • (12)(3) = x2(9) • 36 = 9x2 • 4 = x2

11. practice • If y varies inversely as x and y = 5 when x = 12, find x when y = 3

12. Graphing an inverse variation • Write an inverse variation relating x and y when y = 8 and x = 3. Then graph it. • Find k • xy = k • (3)(8) = 24 • k = 24 so xy=24 y = 24/x • Make a table of values and plot the points

13. practice • Write an inverse variation relating x and y when x = 2 and y = 6. Then graph the relationship.

14. Investigation 7 • Comparing direct and inverse variation

15. compare • Direct variation is a relationship between 2 variables whose ratio is constant. The equation y = kx, where k is a nonzero constant called the constant of variation, shows direct variation between variables x and y. • Identify the constant of variation, given that y varies directly with x • y is 10 when x = 2 • y is 3 when x = 6

16. Word problem • Alex walks 3 miles per hour. If he walks at that rate for twice as long, he will travel twice as far. The ratio of the distance and time is always the same. • Identify the constant of variation- rate, k = ? • Write an equation of direct variation that relates Alex's time to his distance traveled

17. Inverse variation • In inverse variation, as x increases, y decreases. An inverse variation describes a relationship between 2 variables whose product is a constant. The equation xy=k , where k is a nonzero constant, defines an inverse variation between x and y • Identify the constant of variation, given that y varies inversely with x. then write the constant of variation. • y is 1 when x is 3 • y is 4 , when x is 1/2

18. Inverse variation • The equation of inverse variation can be written as y = k/x or xy = k

19. Word problem • Alex lives 4 miles from school. If he walks at a slower rate than normal, it will take him longer to reach his destination. In other words, the more time he spends walking home, the slower he is actually walking. This represents an inverse variation. • Identify the constant of variation. • Write an equation of inverse variation that relates Alex's time to his rate of speed.

20. graphing • A direct variation graph is linear • An inverse variation graph is not linear and never intersects the x-axis