9-1/2-3 NotesDirect and Inverse Variation
Direct VariationY 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:
1 2
1 2
y y
x x
x y5 115 325 5
D P100 642,640 ?
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.100 64
2640100 168960
1689.6
xx
x
x y5 115 325 5
D P100 642,640 ?
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.100 64
2640100 168960
1689.6
xx
x
Do numbers 1 to 4…Compare answers as a class.
Key #1-41. Yes, x and y vary directly, because
2.
3. y = -64. y = 56/3
15 30 15 30 55
3 6 3 6 1
1
1
5y
kx
1
1
5.4 27
3.8 19
yk
x
Inverse VariationY 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
ky
x
1 2
2 1
x x
y y
1 1 2 2 ...x y x y k
x y-1 -6-2 -31 62 3
S T150 5? 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.
150 5
4150 20
7.5
xx
x
x y-1 -6-2 -31 62 3
S T150 5? 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.
150 5
4150 20
7.5
xx
x
Do numbers 5 to 8…Compare answers as a class.
KEY #5-85. Yes, x and y are inversely proportional
because
6. y = 400
7. k = .6 (.4) = .24
8. x = 8
( 4)( 6) ( 3)( 8) 3(8) 4(6)
Joint variationZ 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 5Z 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.56 7 10
560.8
70
k
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.
3
kxz
y
3
83
28
38
3
k
k
k
3
3 6
418
649
32
z
z
z
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!
0.5A bh3
BhV 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 solvefor z:
kxyz
w
5 645
230
452
45 15
3
k
k
k
k
3 4 8
166
z
z
Do numbers 9 to 20…Compare as a class.
Answers.1. yes; k=52. k=27/193. y=-64. y=56/35. yes; k=246. y=4007. k=6/258. x=89. k=-110. z=(0.5y)/x11. directly; k=512. b
13. y=-2214. x=100/715. k=316. 5 hours17. 14 days18. l varies directly
with V and inversely with the product of w and h.
19. z=4/320. k=4186