Types of Variation Direct Variation: y varies directly as x. As x increases, y also increases. As x decreases, y also decreases. Equation for Direct Variation: y = kx Inverse Variation: y varies inversely as x. As x increases, y decreases. As x decreases, y increases. The product of x and y is always constant. Equation for Inverse Variation: xy = k Joint Variation: z varies jointly as x and y. As x and y increases, z also increases. As x and y decrease, z also decreases. Equation for Joint Variation: y = kxz

k is the constant of variation The graph of Direct Variation is a straight line that passes through the origin whose slope, k, is the constant of variation. Let’s look at the graph of y = 2x The graph of y = 2x is a straight line passing through the origin with a slope of 2.

k is the constant of variation The graph of Inverse Variation is a rectangular hyperbola such that the product of x and y is k, the constant of variation. Let’s look at the graph of xy = 2 The graph of xy = 2 is a rectangular hyperbola whose product of x and y is always 2.

Direct Variation Example # 1 If three men earn $180 in one day, how much will 15 men earn at the same rate of pay? First write an equation 180 = k(3) Find the constant of variation 180/3 = k k = 60 Use the constant of variation along with the given information y = 60(15) y = 900 Therefore, 15 men will earn $900 in one day at the same rate of pay That was easy

Direct Variation Example # b If a boat travels 132 miles in 11 hours, how far an it travel in 38.5 hours traveling at the same rate of speed? First write an equation 132 = k(11) Find the constant of variation 132/11 = k k = 12 Use the constant of variation along with the given information y = 12(38.5) y = 462 Therefore, the boat can travel 462 miles traveling at the same rate of speed. That was easy

Inverse Variation Example # a The speed of a gear varies inversely as the number of teeth. If a gear which has 36 teeth makes 30 revolutions per minute, how many revolutions per minute will a gear which has 24 teeth make? First write an equation (30)(36) = k Find the constant of variation k = 1080 Use the constant of variation along with the given information y(24) = 1080 y = 1080/24 y = 45 Therefore, a gear with 24 teeth will make 45 revolutions per minute. That was easy

Inverse Variation Example # 2 If 8 snow plows can plow the airport in 6 hours, how many hours will it take 3 snow plows to plow the same airport? First write an equation (6)(8) = k Find the constant of variation k = 48 Use the constant of variation along with the given information y(3) = 48 y = 48/3 y = 16 Therefore, it will take 3 snow plows 16 hours to plow the airport. That was easy

Joint Variation Example # a Suppose y varies jointly as x and z. Find y when x = 9 and z = 2, if y = 20 when z = 3 and x = 5. Method 1 Method 2 Asi De Facil

Joint Variation Example # 2 Suppose y varies jointly as x and z. Find y when x = 9 and z = -3, if y = -50 when z = 5 and x = -10. Method 1 Method 2 Asi De Facil

Combined Variation # 1 That was easy When one quantity varies directly and inversely as two or more quantities. Suppose f varies directly as g and f varies inversely as h. Find g when f = 18 and h = -3, if g = 24 when h = 2 and f = 6. f varies directly as g so g goes in the numerator. Direct Variation f varies inversely as h so h goes in the denominator. Inverse Variation and and That was easy

Combined Variation # b That was easy When one quantity varies directly and inversely as two or more quantities. Suppose a varies directly as b and a varies inversely as c. Find b when a = 7 and c = -8, if b = 15 when c = 2 and a = 4. a varies directly as b so b goes in the numerator. Direct Variation a varies inversely as c so c goes in the denominator. Inverse Variation and and That was easy

This variation stuff is pretty easy. I feel like jumping for joy! Oh my! I think I’ll just push the easy button. That was easy

Variation Word Problems The volume of a gas varies v varies inversely as the pressure p and directly as the temperature t. Write an equation to represent the volume of a gas in terms of pressure and temperature. Is your equation a direct, joint, inverse, or combined variation? A certain gas has a volume of 8 liters, a temperature of 275 Kelvin, and a pressure of 1.25 atmospheres. If the gas is compressed to a volume of 6 liters and is heated to 300 Kelvin, what will the pressure be?