Warm up
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Warm up
Determine the asymptotes for:
)2(
)3)(2()(
xx
xxxf
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Lesson 3-8 Direct, Inverse & Joint Variation
Objective: To recognize and use direct variation to solve problems
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Definition:
Y varies directly as x means that y = kx where k is the constant of variation.
Another way of writing this is k =
In other words:
* As x increases in value, y increases or
* As x decreases in value, y decreases.
y
x
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X Y 30 10 15 5 9 3
Note: X decreases,
30, 15, 9
And Y decreases.
10, 5, 3
What is the constant of variation of the table above?Since y = kx we can say
Therefore:
10/30=k or k = 1/3 5/15=k or k = 1/3
3/9=k or k =1/3 Note k stays constant.
y = 1/3x is the equation!
yk
x
Examples of Direct Variation:
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Direct Variation
y1
x1
y2
x2
Direct variation uses the following formula:
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Direct Variation
example:
if y varies directly as x and y = 10 as x = 2.4, find x when y =15.
what x and y go together?
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Direct Variation
if y varies directly as x and y = 10 as x = 2.4, find x when y =15
10
2.4
15
x
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Direct Variation
Example:If y varies directly as the square of x and y = 30 when x = 4, find x when y=270.
y=kx2
30=k42
k=1.875270=1.875x2
x=12 12
270
4
3022
xx
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Inverse Variation
Inverse is very similar to direct, but in an inverse relationship as one value goes up, the other goes down.
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Inverse Variation
If y varies inversely as x, then
for some constant k.x
ky
k is still called the constant of variation.
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Inverse VariationWith Direct variation we
Divide our x’s and y’s. In Inverse variation we will
Multiply them.x1y1 = x2y2
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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
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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 = 13.5
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Practice
If t varies inversely as q. If t = 240 when q = 0.01, then find the value of t when q = 8
Two rectangles have the same area. The length of a rectangle varies inversely as the width. If the length of a rectangle is 20 ft when the width is 8 ft, find the length of the rectangle when the width is 10 ft.?
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Joint and Combined Variation
Joint variation is like direct variation but it involves more than one quantity.
Combined variation combines both direct and inverse variation in the same problem.
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Joint and Combined Variation
For example: if z varies jointly with x & y, then z=kxy.
Ex: if y varies inversely with the square of x, then y=k/x2.
Ex: if z varies directly with y and inversely with x, then z=ky/x.
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Example
y varies jointly as x and w and inversely as the square of z. Find the equation of variation when y = 100, x = 2, w = 4, and z = 20. Then find k.
2z
kxwy
5000k
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Example
If y varies jointly as x and z, and y = 12 when x = 9 and z = 3, find z when y = 6 and x = 15.