11-1 Inverse Variation. Determine whether each table or equation represents an inverse or a direct variation. Explain. 14. x y

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1 14. Determine whether each table or equation represents an inverse or a direct variation. Explain. x y The equation is an inverse variation if the products of the two values remain constant. Find xy. The equation is an inverse variation if the products of the two values remain constant. Find xy Notice that xy is not constant. So, the table does not represent an indirect variation. The equation is a direct variation if the quotient of the two values remains constant. Find. Notice that xy is constant, so the table represents the inverse variation xy = 30. Confirm with a graph. x y Inverse; xy = 30. The table of values represents the direct variation. Confirm on a graph. 16. x y esolutions Manual - Powered by Cognero Page 1

2 Direct; y =. Make a table of values. Choose values of x and y that have a product of x y = 0 In a direct variation, y = kx. Because the equation can be written as y = 5x, this equation represents a direct variation. Direct; y = 5x. 20. x = 14y x (x, y) ( 16, 2.5) ( 12, 3.3) 8 5 ( 8, 5) 4 10 ( 4, 10) 0 undefined undefined 4 10 (4, 10) 8 5 (8, 5) (12, 3.3) (16, 2.5) Plot each point and draw a smooth curve that connects the points. Remember that there is a disconnect at x = 0. Connect the points on one side of the y-axis, and then connect the points on the other side of the y-axis, but do not make any connections across the y-axis. In a direct variation, y = kx. Because the equation can be written as, this equation represents a direct variation. Direct; y = kx. Assume that y varies inversely as x. Write an inverse variation equation that relates x and y. Then graph the equation. 22. y = 2 when x = 20 xy = 40 Find the constant of variation. The constant of variation is 40. So, an equation that 24. y = 6 when x = 3 relates x and y is xy = 40 or. Find the constant of variation. esolutions Manual - Powered by Cognero Page 2

3 The constant of variation is 18. So, an equation that relates x and y is xy = 18 or. Make a table of values. Choose values of x and y that have a product of 18. x (x, y) ( 12, 1.5) 9 2 ( 9, 2) 6 3 ( 6, 3) 3 6 ( 3, 6) 0 undefined undefined 3 6 (3, 6) 6 3 (6, 3) 9 2 (9, 2) (12, 1.5) Plot each point and draw a smooth curve that connects the points. Remember that there is a disconnect at x = 0. Connect the points on one side of the y-axis, and then connect the points on the other side of the y-axis, but do not make any connections across the y-axis. xy = y = 4 when x = 16 Find the constant of variation. The constant of variation is 64. So, an equation that relates x and y is xy = 64 or. Make a table of values. Choose values of x and y that have a product of 64. x (x, y) 16 4 ( 16, 4) ( 12, 5.3) 8 8 ( 8, 8) 4 16 ( 4, 16) 0 undefined undefined 4 16 (4, 16) 8 8 (8, 8) (12, 5.3) 16 4 (16, 4) Plot each point and draw a smooth curve that connects the points. Remember that there is a disconnect at x = 0. Connect the points on one side of the y-axis, and then connect the points on the other side of the y-axis, but do not make any connections across the y-axis. esolutions Manual - Powered by Cognero Page 3

4 30. If y = 4 when x = 14, find x when y = 5. Let x 1 = 14, y 1 = 4, and y 2 = 5. Solve for x 2. xy = 64 So, when y = 5, x = If y = 15 when x = 2, find y when x = 3. Solve. Assume that y varies inversely as x. 28. If y = 12 when x = 3, find x when y = 6. Let x 1 = 2, y 1 = 15, and x 2 = 3. Solve for y 2. Let x 1 = 3, y 1 = 12, and y 2 = 6. Solve for x 2. So, when x = 3, y = So, when y = 6, x = 6. 6 esolutions Manual - Powered by Cognero Page 4

5 34. EARTH SCIENCE The water level in a river varies inversely with air temperature. When the air temperature was 90 Fahrenheit, the water level was 11 feet. If the air temperature was 110 Fahrenheit, what was the level of water in the river? Let x 1 = 11, y 1 = 90, and y 2 = 110. Solve for x Nicole earns $14 for babysitting 2 hours, and $21 for babysitting 3 hours. "Nicole earns $14 for babysitting 2 hours, and $21 for babysitting 3 hours." is an example of direct variation. The number of hours times the rate per hour equals the total pay. Consider the following scenarios in a table. # of hrs total pay So, when the air temperature was 110 Fahrenheit, the level of water in the river was 9 ft. 9 ft Determine whether each situation is an example of an inverse or a direct variation. Justify your reasoning. 36. The drama club can afford to purchase 10 wigs at $2 each or 5 wigs at $4 each. This situation is an example of inverse variation. In an inverse variation, xy equals a constant k. The cost per wig times the number of wigs is constant Thus, the ratio is a constant $7. Direct; the number of hours times the rate per hour equals the total pay. The ratio constant $7. is a The total amount the drama club can spend is a constant, $20. As the cost per wig increases, the number of wigs they can buy decreases. Inverse; the cost per wig times the number of wigs equals the total amount they can spend, $20. esolutions Manual - Powered by Cognero Page 5

6 40. Determine whether each table or graph represents an inverse or a direct variation. Explain. x y The equation is an inverse variation if the products of the two values remain constant. Find xy Notice that xy is not constant. So, the table does not represent an inverse variation. The equation is a direct variation if the quotient of the two values remains constant. Find. 42. Identify 2 points on the curve. (1,2) and (2, 1). Find xy for each point Then, the constant of variation is 2. Thus, this is an example of inverse variation, xy = 2. Inverse; xy = 2. x y The table of values represents the direct variation y = 0.2x. Confirm on a graph. Direct; y = 0.2x. esolutions Manual - Powered by Cognero Page 6