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Warm Up

Preview. Warm Up. California Standards. Lesson Presentation. Warm Up Solve each proportion. 1. 2. 4.2. 10. 3. 4. 2.625. 2.5. 5. The value of y varies directly with x , and y = – 6 when x = 3. Find y when x = – 4. 8.

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Warm Up

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Solve each proportion. 1. 2. 4.2 10 3. 4. 2.625 2.5 5. The value of y varies directly with x, and y = –6 when x = 3. Find y when x = –4. 8 6. The value of y varies directly with x, and y = 6 when x = 30. Find y when x = 45. 9

  3. California Standards Preparation for 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. Also covered: 17.0

  4. Vocabulary inverse variation

  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. 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. 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.

  7. Can write in y = form. Additional Example 1A: Identifying an Inverse Variation Tell whether the relationship is an inverse variation. Explain. Method 1 Write a function rule. 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.

  8. Cannot write in y = form. Additional Example 1B: Identifying an Inverse Variation Tell whether the relationship is an inverse variation. Explain. Method 1 Write a function rule. y = 5x The relationship is not an inverse variation. 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.

  9. xy equals the constant 14, so the relationship is an inverse variation. The equation can be written in the form y = Additional Example 1C: Identifying an Inverse Variation Tell whether the 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.

  10. Cannot write in y = form. Check It Out! Example 1a Tell whether each relationship is an inverse variation. Explain. Method 1 Write a function rule. y = –2x The relationship is not an inverse variation. 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.

  11. Can write in y = form. Check It Out! Example 1b Tell whether each relationship is an inverse variation. Explain. Method 1 Write a function rule. 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.

  12. Cannot write in y = form. Check It Out! Example 1c Tell whether each relationship is an inverse variation. Explain. 2x + y = 10 The relationship is not an inverse variation.

  13. Helpful Hint Since k is a nonzero constant, xy ≠ 0. Therefore, neither x nor y can equal 0, and the graph will not intercept the x- or y-axes.

  14. 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). In other words, (0, 0) can never be a solution of an inverse variation equation.

  15. Additional 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 Write the rule for constant of variation. = –12(0.5) 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.

  16. x 1 2 4 –1 0 –4 –2 y 1.5 3 6 –1.5 –6 –3 undef. Additional 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.

  17. ● ● ● ● ● ● Additional 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.

  18. = 10 Substitute 10 for x and for y. Check It Out! Example 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 Step 2 Use the value of k to write an inverse variation equation. Write the rule for inverse variation. Substitute 5 for k.

  19. Check It Out! Example 2 Continued Write and graph the inverse variation in which y = when x = 10. Step 3 Use the equation to make a table of values.

  20. ● ● ● ● ● ● Check It Out! Example 2 Continued Write and graph the inverse variation in which y = when x = 10. Step 4 Plot the points and connect them with smooth curves.

  21. Additional Example 3: Transportation Application The inverse variation xy = 350 relates the constant speed x in mi/h to the time y in hours that it takes to travel 350 miles. Determine a reasonable domain and range and then graph this inverse variation. Step 1 Solve the function for y. xy = 350 Divide both sides by x.

  22. x 20 40 60 80 y 17.5 8.75 5.83 4.38 Additional Example 3 Continued Step 2 Decide on a reasonable domain and range. Speed is never negative and x ≠ 0 x > 0 Because x and xy are both positive, y is also positive. y > 0 Step 3 Use values of the domain to generate reasonable ordered pairs.

  23. ● ● ● ● Additional Example 3 Continued Step 4 Plot the points. Connect them with a smooth curve.

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

  25. Check It Out! Example 3 The inverse variation xy = 100 represents the relationship between the pressure x in atmospheres (atm) and the volume y in mm3 of a certain gas. Determine a reasonable domain and range and then graph this inverse variation. Step 1 Solve the function for y. xy = 100 Divide both sides by x.

  26. x 10 20 30 40 y 10 5 3.34 2.5 Check It Out! Example 3 Continued Step 2 Decide on a reasonable domain and range. Pressure is never negative and x ≠ 0 x > 0 Because x and xy are both positive, y is also positive. y > 0 Step 3 Use values of the domain to generate reasonable pairs.

  27. ● ● ● ● Check It Out! Example 3 Continued Step 4 Plot the points. Connect them with a smooth curve.

  28. 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.

  29. Let and Let y vary inversely as x. Find Substitute 5 for 3 for and 10 for . Solve for by dividing both sides by 5. Additional Example 4: Using the Product Rule Write the Product Rule for Inverse Variation. Simplify. Simplify.

  30. Let and Let y vary inversely as x. Find Substitute 2 for –4 for and –6 for Solve for by dividing both sides by –4. Check It Out! Example 4 Write the Product Rule for Inverse Variation. Simplify. Simplify.

  31. Additional Example 5: Physical Science Application Boyle’s law states that the pressure of a quantity of gas x varies inversely as the volume of the gas y. The volume of gas inside a container is 400 in3 and the pressure is 25 psi. What is the pressure when the volume is compressed to 125 in3? Use the Product Rule for Inverse Variation. Substitute 400 for x1, 125 for x2, and 25 for y1. (400)(25) = (125)y2 Simplify. Solve for y2 by dividing both sides by 125.

  32. When the gas is compressed to 125 in3, the pressure increases to 80 psi. Additional Example 5 Continued Boyle’s law states that the pressure of a quantity of gas x varies inversely as the volume of the gas y. The volume of gas inside a container is 400 in3 and the pressure is 25 psi. What is the pressure when the volume is compressed to 125 in3?

  33. Check It Out! Example 5 On a balanced lever, weight varies inversely as the distance from the fulcrum to the weight. The diagram shows a balanced lever. How much does the child weigh?

  34. Substitute 3.2 for , 60 for and 4.3 for Solve for by dividing both sides by 3.2. Check It Out! Example 5 Continued Use the Product Rule for Inverse Variation. Simplify. Simplify. The child weighs 80.625 lb.

  35. Lesson Quiz: Part I 1. Write and graph the inverse variation in which y = 0.25 when x = 12.

  36. Lesson Quiz: Part II 2. The inverse variation xy = 210 relates the length y in cm to the width x in cm of a rectangle with an area of 210 cm2. Determine a reasonable domain and range and then graph this inverse variation.

  37. Lesson Quiz: Part III 3. Let x1= 12, y1 = –4, and y2 = 6, and let y vary inversely as x. Find x2. –8

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