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Direct and Inverse Variation: Understanding the Relationship Between Variables

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  1. 9-1/2-3 Notes Direct and Inverse Variation

  2. 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:

  3. 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. To solve for the missing value, set up a proportion…solve by cross multiplying. 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.

  4. 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. To solve for the missing value, set up a proportion…solve by cross multiplying. 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.

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

  6. Key #1-4 • 1. Yes, x and y vary directly, because • 2. • 3. y = -6 • 4. y = 56/3

  7. 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

  8. 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. To solve for x, set up an inverse proportion and cross multiply. 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.

  9. 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. To solve for x, set up an inverse proportion and cross multiply. 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.

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

  11. KEY #5-8 • 5. Yes, x and y are inversely proportional because • 6. y = 400 • 7. k = .6 (.4) = .24 • 8. x = 8

  12. 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.

  13. 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.

  14. 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. • 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.

  15. 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.

  16. 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:

  17. Do numbers 9 to 20…Compare as a class.

  18. Answers. • 1. yes; k=5 • 2. k=27/19 • 3. y=-6 • 4. y=56/3 • 5. yes; k=24 • 6. y=400 • 7. k=6/25 • 8. x=8 • 9. k=-1 • 10. z=(0.5y)/x • 11. directly; k=5 • 12. b • 13. y=-22 • 14. x=100/7 • 15. k=3 • 16. 5 hours • 17. 14 days • 18. l varies directly with V and inversely with the product of w and h. • 19. z=4/3 • 20. k=4186

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