Inverse Variation

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.

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.

Example 1: 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.

Example 2: 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.

Example 3: 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.

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. Example 4

Tell whether each relationship is an inverse variation. Explain. Example 5 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.

2x + y = 10 Tell whether each relationship is an inverse variation. Explain. Example 6 Cannot write in y = form. The relationship is not an inverse variation.

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.

Example 7: 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.

Example 7 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– undef. –6–6 –3–3 – 1.5

Example 7 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. ● ● ● ● ● ●

Example 8 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.

Write and graph the inverse variation in which y = when x = 10. Step 3 Use the equation to make a table of values. Example 8 Continued x –4–2–10124 y –1.25–2.5–5undef

Example 8 Continued Step 4 Plot the points and connect them with smooth curves. Write and graph the inverse variation in which y = when x = 10. ● ● ● ● ● ●

Example 9: 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. Use the graph to estimate how long it will take to travel 350 miles driving 55 mi/h. Step 1 Solve the function for y so you can graph it. xy = 350 Divide both sides by x.

Example 9 Continued Step 2 Decide on a reasonable domain and range. x > 0 y > 0 Length 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 ordered pairs y x

Example 9 Continued Step 4 Plot the points. Connect them with a smooth curve. ● ● ● ● Step 5 Find the y-value where x = 55. When the speed is 55 mi/h, the travel time is about 6 hours.

Example 10 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.

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 y x Example 10 Continued

Step 4 Plot the points. Connect them with a smooth curve. Example 10 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. ● ● ● ●

Example 11: Using the Product Rule Let and Let y vary inversely as x. Find Write the Product Rule for Inverse Variation. Substitute 5 for 3 for and 10 for. Simplify. Solve for by dividing both sides by 5. Simplify.

Example 12 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.

Example 13: Physics 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 in 3 and the pressure is 25 psi. What is the pressure when the volume is compressed to 125 in 3 ? Use the Product Rule for Inverse Variation. Substitute 400 for 125 for and 25 for Simplify. Solve for by dividing both sides by 125. (400)(25) = (125)y 2

Example 13 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 in 3 and the pressure is 25 psi. What is the pressure when the volume is compressed to 125 in 3 ? When the gas is compressed to 125 in 3, the pressure increases to 80 psi.

Example 14 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?

Example 14 Continued Use the Product Rule for Inverse Variation. Substitute 3.2 for, 60 for and 4.3 for Simplify. Solve for by dividing both sides by 3.2. Simplify. The child weighs lb.

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

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 cm 2. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate the length when the width is 14 cm.

Lesson Quiz: Part III 3. Let x 1 = 12, y 1 = – 4, and y 2 = 6, and let y vary inversely as x. Find x 2. –8–8