Direct and Inverse Variation

Direct Variation Y varies directly to x, when x and y are related by the equation: y=kx. Here k is the constant of variation. In other words…generally as x increases y will also increase and vice versa. Also, x and y are directly proportional so:

xy DP ,640? Ex1: Determine from the table if y varies directly to x. If so, find the constant of variation. Ex2: D varies directly with P. Find the missing value. X and y are directly related because if you plug into the equation y=kx, you will get the same k value each time…k=1/5. To solve for the missing value, set up a proportion…solve by cross multiplying.

xy DP ,640? Ex1: Determine from the table if y varies directly to x. If so, find the constant of variation. Ex2: D varies directly with P. Find the missing value. X and y are directly related because if you plug into the equation y=kx, you will get the same k value each time…k=1/5. To solve for the missing value, set up a proportion…solve by cross multiplying.

Do numbers 1 to 4…Compare answers as a class.

Key # Yes, x and y vary directly, because y = y = 56/3

Inverse Variation Y varies inversely to x, when x and y are related by the equation In other words…generally as x increases y will decrease and vise versa. Also x and y are inversely proportional so: (basically the x1 and y1 are diagonally across from each other, as are the x2 and y2) This also means that

xy ST 1505 ?4 Ex3: Determine from the table if y is inversely proportional to x. If so, find the constant of variation. Ex4: S varies inversely with T. Find the missing value. X and y are inversely related because when plugged into the equation y=k/x, you get the same thing every time for k. k=6. To solve for x, set up an inverse proportion and cross multiply.

xy ST 1505 ?4 Ex3: Determine from the table if y is inversely proportional to x. If so, find the constant of variation. Ex4: S varies inversely with T. Find the missing value. X and y are inversely related because when plugged into the equation y=k/x, you get the same thing every time for k. k=6. To solve for x, set up an inverse proportion and cross multiply.

Do numbers 5 to 8…Compare answers as a class.

KEY # Yes, x and y are inversely proportional because 6. y = k =.6 (.4) = x = 8

Joint variation Z varies jointly with x and y, when x, y, and z are related by the equation z=kxy. ‘Varies’ tells you where to put the equal sign. k always comes after the equal sign.

Example 5 Z varies jointly as x and y, if z=56 when x=7 and y=10, find the constant of variation. To solve use the joint variation equation z=kxy and solve for k.

To solve you need to make an equation that relates x, y, and z. Remember the varies tells you where to put the equal sign, k always comes after the equal, directly means multiply, and inversely means divide. This means your equation should be: Plug in the values they give for x, y, and z to solve for k. Use this k value to solve for z when x=6 and y=4. Example 6 Z varies directly with x and inversely with the cube of y. When x=8 and y=2, z=3. Find z when x=6 and y=4.

Example 7 Describe the variation that is modeled by each formula. Remember the equal sign is represented by varies when you are describing a variation! If you are describing a variable that is multiplied you will say directly. If you are describing a variable the is divided you will say inversely. If you are describing two variables that are both multiplied say jointly. If there is a number then it is the constant of variation! A varies jointly with b and h, when 0.5 is the constant of variation. V varies jointly with B and h, when 1/3 is the constant of variation.

Example 8 z varies jointly with x and y and inversely with w. When x = 5, y = 6, and w = 2, z = 45. Write a function that models this relationship, then find z when x = 4, y = 8, and w = 16. To solve you need to make an equation that relates x, y, and z. Remember the varies tells you where to put the equal sign, k always comes after the equal, directly means multiply, inversely means divide, jointly means multiply by both variables. This means your equation should be: Then use the first set of values to solve for the k value: Then plug in the second set of values with k to solve for z:

Do numbers 9 to 20…Compare as a class.

Answers. 1. yes; k=5 2. k=27/19 3. y=-6 4. y=56/3 5. yes; k=24 6. y= k=6/25 8. x=8 9. k= z=(0.5y)/x 11. directly; k=5 12. b 13. y= x=100/7 15. k= hours days 18. l varies directly with V and inversely with the product of w and h. 19. z=4/3 20. k=4186