12.1 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Model Inverse Variation

12.1 Warm-Up 1. Rewrite 2x – 5y = 0 as a direct variation equation. 2. The number of words w you type on a keyboard varies directly with the number of minutes m that you type. You can type 360 words in 8 minutes. How many words can you type in 12 minutes? ANSWER 540 words ANSWER y = x 2 5

12.1 Example 1 Tell whether the equation represents direct variation, inverse variation, or neither. a. xy = 4 SOLUTION Write original equation. xy = 4 Divide each side by x. y = 4 x Because xy = 4 can be written in the form y =, xy = 4 represents inverse variation. The constant of variation is 4. a x b. = x y 2 c. y = 2x + 3

12.1 Example 1 SOLUTION b. = x y 2 = x y 2 Write original equation. Multiply each side by 2. y = 2x Because = x can be written in the form y = ax, = x represents direct variation. y 2 y 2

12.1 Example 1 SOLUTION c. y = 2x + 3 Because y = 2x + 3 cannot be written in the form y = or y = ax, y = 2x + 3 does not represent either direct variation or inverse variation. a x

12.1 Guided Practice Tell whether the equation represents direct variation, inverse variation, or neither. 1. y = 2x2x ANSWER inverse variation 2. 4y = 3x ANSWER direct variation 3. 5x – y = 3 ANSWER neither 4. xy = 1212 ANSWER inverse variation

12.1 Example 2 STEP 1 Make a table by choosing several integer values of x and finding the values of y. Then plot the points. To see how the function behaves for values of x very close to 0 and very far from 0, make a second table for such values and plot the points. Graph y =. 4 x SOLUTION

12.1 Example 2 STEP 2 Connect the points in Quadrant I by drawing a smooth curve through them. Repeat for the points in Quadrant III.

12.1 Example 3 Graph y =. Then find the domain and range of the function. – 4 x SOLUTION Notice that. So, for every nonzero value of x, the value of y in is the opposite of the value of y in. You can graph by reflecting the graph of (see Example 2 ) in the x -axis. Both the domain and the range of the function are all real numbers except 0. y – 4 x = –1 = 4 x y – 4 x = y 4 x = – y 4 x = y 4 x =

12.1 Guided Practice 5. Graph (a) y = and (b) y =. x 3 – 3 x

12.1 Example 4 The variables x and y vary inversely, and y = 6 when x = – 3. a. Write an inverse variation equation that relates x and y. b. Find the value of y when x = 4. SOLUTION Use the fact that x = – 3 and y = 6 to find the value of a. a y x =. Because y varies inversely with x, the equation has the form a.

12.1 Example 4 An equation that relates x and y is y = –18 x. b. When x = 4, y = –18 4 = – 2. 9 Write inverse variation equation. y a x = Substitute – 3 for x and 6 for y. 6 a –3 = Multiply each side by – 3. – 18 = a

12.1 Guided Practice The variables x and y vary inversely, and y = –2 when x = 12. Write an inverse variation equation that relates x and y. Then find the value of y when x = – 3. y = –24 x ; 8 ANSWER

12.1 Example 5 SOLUTION Tell whether the table represents inverse variation. If so, write the inverse variation equation. Find the products xy for all pairs (x, y): –5(2.4) = –12 –3(4) = –12 4(–3) = –12 8(–1.5) = –12 24(–0.5) = –12

12.1 Example 5 The products are equal to the same number, –12. So, y inversely with x. ANSWER The inverse variation equation is xy = –12, or y =. –12 x

12.1 Guided Practice 7. Tell whether the ordered pairs (–5, 2), (–4, 2.5), (8, –1.25), and (20, –0.5) represent inverse variation. If so, write the inverse variation equation. inverse variation; y = –10 x ANSWER

12.1 Example 6 A theater company plans to hire people to build a stage set. The work time t (in hours per person) varies inversely with the number p of people hired. The company estimates that 25 people working for 300 hours each can complete the job. Find the work time per person if the company hires 30 people. Theater

12.1 Example 6 SOLUTION STEP 1 Write the inverse variation equation that relates p and t. Write inverse variation equation. t = a p Substitute 25 for p and 300 for t = a Multiply each side by = a 7500 The inverse variation equation is t =. p

12.1 Example 6 STEP p Find t when p = 30: t = = = 250. ANSWER If 30 people are hired, the work time per person is 250 hours.

12.1 Guided Practice 8. WHAT IF? In Example 6, suppose the theater company estimates that 20 people working for 270 hours each can complete the job. Find the work time per person if the company hires 30 people. ANSWER If 30 people are hired, the work time per person is 180 hours.

12.1 Lesson Quiz Graph y = 1. – 6 x ANSWER

12.1 Lesson Quiz 3. x = 12, y = 7 Given that y varies with x, use the specified values to write an inverse variation equation that relates x and y. Then find the values of y when x = 3. ANSWER y = – ; – x 2. x = – 6, y = 5 ANSWER y = ; x