1. Warm Up Solve each proportion. 1. 2. 3. 4. 5. The value of y varies directly with x, and y = – 6 when x = 3. Find y when x = – 4. 6. The value of y varies directly with x, and y = 6 when x = 30. Find y when x = 45. 10 4.2 2.6252.5 8 9

2. Inverse Variation

3. Identify, write, and graph inverse operations. Objective

4. inverse variation Vocabulary

5. A relationship that can be written in the form y =, where k is a nonzero constant and x ≠ 0, is an inverse variation. The constant k is the constant of variation. Inverse variation implies that one quantity will increase while the other quantity will decrease (the inverse, or opposite, of increase). Multiplying both sides of y = by x gives xy = k. So, for any inverse variation, the product of x and y is a nonzero constant.

6. A direct variation is an equation that can be written in the form y = kx, where k is a nonzero constant. Remember!

7. There are two methods to determine whether a relationship between data is an inverse variation. You can write a function rule in y = form, or you can check whether xy is a constant for each ordered pair.

8. Example 1A: Identifying an Inverse Variation Tell whether each relationship is an inverse variation. Explain. Method 1 Write a function rule. Can write in y = form. The relationship is an inverse variation. Method 2 Find xy for each ordered pair. 1(30) = 30, 2(15) = 30, 3(10) = 30 The product xy is constant, so the relationship is an inverse variation.

9. Example 1B: Identifying an Inverse Variation Tell whether each relationship is an inverse variation. Explain. Method 1 Write a function rule. Cannot write in y = form. The relationship is not an inverse variation. y = 5x Method 2 Find xy for each ordered pair. 1(5) = 5, 2(10) = 20, 4(20) = 80 The product xy is not constant, so the relationship is not an inverse variation.

10. Example 1C: Identifying an Inverse Variation Tell whether each relationship is an inverse variation. Explain. 2xy = 28 Find xy. Since xy is multiplied by 2, divide both sides by 2 to undo the multiplication. xy = 14 Simplify. xy equals the constant 14, so the relationship is an inverse variation.

11. Tell whether each relationship is an inverse variation. Explain. Method 1 Write a function rule. Cannot write in y = form. The relationship is not an inverse variation. y = – 2x Method 2 Find xy for each ordered pair. – 12 (24) = – 228, 1( – 2) = – 2, 8( – 16) = – 128 The product xy is not constant, so the relationship is not an inverse variation. Try 1a

12. Tell whether each relationship is an inverse variation. Explain. Try 1b Method 1 Write a function rule. Can write in y = form. The relationship is an inverse variation. Method 2 Find xy for each ordered pair. 3(3) = 9, 9(1) = 9, 18(0.5) = 9 The product xy is constant, so the relationship is an inverse variation.

13. 2x + y = 10 Tell whether each relationship is an inverse variation. Explain. Try 1c Cannot write in y = form. The relationship is not an inverse variation.

14. Since k is a nonzero constant, xy ≠ 0. Therefore, neither x nor y can equal 0, and no solution points will be on the x- or y-axes. Helpful Hint

15. An inverse variation can also be identified by its graph. Some inverse variation graphs are shown. Notice that each graph has two parts that are not connected. Also notice that none of the graphs contain (0, 0). This is because (0, 0) can never be a solution of an inverse variation equation.

16. Example 2: Graphing an Inverse Variation Write and graph the inverse variation in which y = 0.5 when x = – 12. Step 1 Find k. k = xy = – 12(0.5) Write the rule for constant of variation. Substitute –12 for x and 0.5 for y. = – 6 Step 2 Use the value of k to write an inverse variation equation. Write the rule for inverse variation. Substitute –6 for k.

17. Example 2 Continued Write and graph the inverse variation in which y = 0.5 when x = – 12. Step 3 Use the equation to make a table of values. y –2 –4 x –1–1 0 124 1.53 6 undef. –6–6 –3–3 – 1.5

18. Example 2 Continued Write and graph the inverse variation in which y = 0.5 when x = – 12. Step 4 Plot the points and connect them with smooth curves. ● ● ● ● ● ●

19. Try 2 Write and graph the inverse variation in which y = when x = 10. Step 1 Find k. k = xy Write the rule for constant of variation. = 5 Substitute 10 for x and for y. = 10 Step 2 Use the value of k to write an inverse variation equation. Write the rule for inverse variation. Substitute 5 for k.

20. Write and graph the inverse variation in which y = when x = 10. Step 3 Use the equation to make a table of values. Try 2 Continued x –4–2–10124 y –1.25–2.5–5undef.52.51.25

21. Try 2 Continued Step 4 Plot the points and connect them with smooth curves. Write and graph the inverse variation in which y = when x = 10. ● ● ● ● ● ●

22. Recall that sometimes domain and range are restricted in real-world situations. Remember!

23. Example 3 The inverse variation xy = 100 represents the relationship between the pressure x in atmospheres (atm) and the volume y in mm ³ of a certain gas. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate the volume of the gas when the pressure is 40 atmospheric units. Step 1 Solve the function for y so you can graph it. xy = 100 Divide both sides by x.

24. Step 2 Decide on a reasonable domain and range. x > 0 y > 0 Pressure is never negative and x ≠ 0 Because x and xy are both positive, y is also positive. Step 3 Use values of the domain to generate reasonable pairs. 2.53.34510y 40302010x Example 3 Continued

25. Step 4 Plot the points. Connect them with a smooth curve. Example 3 Continued Step 5 Find the y-value where x = 40. When the pressure is 40 atm, the volume of gas is about 2.5 mm 3. ● ● ● ●

26. The fact that xy = k is the same for every ordered pair in any inverse variation can help you find missing values in the relationship.

28. Example 4: Using the Product Rule Let and Let y vary inversely as x. Find Write the Product Rule for Inverse Variation. Simplify. Solve for by dividing both sides by 5. Simplify.

29. Try 4 Write the Product Rule for Inverse Variation. Simplify. Substitute 2 for –4 for and –6 for Let and Let y vary inversely as x. Find Solve for by dividing both sides by –4.