Topic: Inverse Variation HW: Worksheet Math CC7/8 – April 3 Things Needed Today (TNT): Pencil/Math Notebook/Calculators TwMM 3.2 Math Notebook: Topic: Inverse Variation HW: Worksheet

What’s Happening Today? Warm Up Inverse Variation Inverse Variation Foldable

You will be able to identify, write, & graph an equation of Inverse variation!

What strategies will you use to find the equation? Warm Up Find the equation of the line that passes through the given points. What strategies will you use to find the equation? (1, 4) and (-2, -2) y = 2x + 2

Note: parallel lines have the SAME slope (m) Warm Up Find an equation for the line that satisfies the conditions. Note: parallel lines have the SAME slope (m)

There is not necessarily a constant rate. Inverse Variation Inverse Variation is very similar to direct, BUT in an inverse relationship as one value goes up, the other goes down. There is not necessarily a constant rate.

Inverse variation is a relationship between two variables that can be written in the form y = k/x or xy = k, where k is a nonzero constant and x ≠ 0. In an inverse variation, the product of x and y is constant. k is the constant of proportionality

Inverse Variation x1y1 = x2y2 With Direct variation we Divide our x’s and y’s. In Inverse variation we will Multiply our x’s & y’s. x1y1 = x2y2

How to read Inverse Variation You can read inverse variation as “y varies inversely with x”.

If y varies inversely with x and y = 12 when x = 2, find y when x = 8. Inverse Variation If y varies inversely with x and y = 12 when x = 2, find y when x = 8. x1y1 = x2y2 2(12) = 8y 24 = 8y y = 3

Inverse Variation If y varies inversely as x and x = 18 when y = 6, find y when x = 8. 18(6) = 8y 108 = 8y y = 27 / 2

What to do if you have a table? Inverse Variation The product for xy is constant, so the relationship is an inverse variation with k = 24.

5(80) = 400 7(75)= 525 9(70) = 630 The product for xy is not constant, so the relationship is not an inverse variation.

Word Problem Practice: Inverse Deviation David is building a rectangular flowerbed. He has soil to cover 48 square feet. The flowerbed can be 4, 6, or 12 feet long. For each length x, find the width of the flowerbed y to use all the soil. The area A of the flowerbed is a constant k. The length x times the width y must equal the area, 48. The equation xy = 48 is an inverse variation.

An inverse variation can also be identified by its graph An inverse variation can also be identified by its graph. Since k is a nonzero constant, ≠ 0. Therefore, neither x nor y can equal 0, and no solution points will be on the x-axis or y-axis.

Identify Inverse Variation Tell whether the relationship is an inverse variation. The table shows how 24 cookies can be divided equally among different numbers of students. Number of Students 2 3 4 6 8 Number of Cookies 12 2(12) = 24; 3(8) = 24; 4(6) = 24; 6(4) = 24; 8(3) = 24 xy = 24 The product is always the same. The relationship is an inverse variation: y = . 24 x

Tell whether the relationship is an inverse variation. Try This: Inverse Variation Tell whether the relationship is an inverse variation. x y 2 3 4 5 6 0(2) = 0; 0(3) = 0; 0(4) = 0; 0(5) = 0; 0(6) = 0 xy = 0 The product is always the same. The relationship is an inverse variation: y = . x

Identify Inverse Variation The product is not always the same. Tell whether each relationship is an inverse variation. The table shows the number of cookies that have been baked at different times. Number of Students 12 24 36 48 60 Time (min) 15 30 45 75 The product is not always the same. 12(15) = 180; 24(30) = 720 The relationship is not an inverse variation.

Inverse Variation 2(4) = 8; 2(6) = 12 x 2 4 8 1 y 6 Tell whether the relationship is an inverse variation. x 2 4 8 1 y 6 The product is not always the same. 2(4) = 8; 2(6) = 12 The relationship is not an inverse variation.

Graphing Inverse Variations Graph the inverse variation function. f(x) = 4 x x y –4 –2 –1 1 2 4 –1 –2 –4 – 12 –8 12 8 4 2 1

Graph the inverse variation function. A. f(x) = – 4 x x y Try This: Example 2A Graph the inverse variation function. A. f(x) = – 4 x x y –4 –2 –1 1 2 4 1 2 4 – 12 8 12 –8 –4 –2 –1

Additional Example 2B: Graphing Inverse Variations Graph the inverse variation function. B. f(x) = –1 x x y –3 –2 –1 1 2 3 1 3 1 2 1 – 12 2 12 –2 –1 1 2 – 1 3 –

Try This: Example 2B Graph the inverse variation function. B. f(x) = 8 x x y –8 –4 –2 –1 1 2 4 8 –1 –2 –4 –8 8 4 2 1

Volume of Gas by Pressure on Gas Additional Example 3: Application As the pressure on the gas in a balloon changes, the volume of the gas changes. Find the inverse variation function and use it to find the resulting volume when the pressure is 30 lb/in2. Volume of Gas by Pressure on Gas Pressure (lb/in2) 5 10 15 20 Volume (in3) 300 150 100 75 You can see from the table that xy = 5(300) = 1500, so y = . 1500 x If the pressure on the gas is 30 lb/in2, then the volume of the gas will be y = 1500 ÷ 30 = 50 in3.

Number of Students by Cost per Student Try This: Example 3 An eighth grade class is renting a bus for a field trip. The more students participating, the less each student will have to pay. Find the inverse variation function, and use it to find the amount of money each student will have to pay if 50 students participate. Number of Students by Cost per Student Students 10 20 25 40 Cost per student 8 5 You can see from the table that xy = 10(20) = 200, so y = . 200 x If 50 students go on the field trip, the price per student will be y = 200  50 = $4.

Lesson Quiz: Part 1 Tell whether each relationship is an inverse variation. 1. 2. yes no

Review Inverse Variation y varies inversely as x if such that xy=k or Just as with direct variation, a proportion can be set up solve problems of indirect variation.

A general form of the proportion Lets do an example that can be solved by using the equation and the proportion.

Find y when x=15, if y varies inversely as x and x=10 when y=12 Solve by equation:

Solve by proportion:

Solve this problem using either method. Find x when y=27, if y varies inversely as x and x=9 when y=45. Answer: 15

Lets apply what we have learned. The pressure P of a compressed gas is inversely proportional to its volume V according to Boyle’s Law. A pressure of 40 pounds per square inch is created by 600 cubic inches of a certain gas. Find the pressure when the gas is compressed to 200 cubic inches.

Step #1: Set up a proportion.

Now try this one on your own. A pressure of 20 pounds per inch squared is exerted by 400 inches cubed of a certain gas. Use Boyle’s Law to find the pressure of the gas when it is compressed to a volume of 100 inches cubed.

What does the graph of xy=k look like? Let k=5 and graph.

This is a graph of a hyperbola. Notice: That in the graph, as the x values increase the y values decrease. also As the x values decrease the y values increase.