Inverse Variation

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Inverse Variation

• What is it and how do I know when I see it?

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Inverse Variation
When we talk about an inverse variation, we are
talking about a relationship where as x
increases, y decreases or x decreases, y
increases at a CONSTANT RATE.
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Definition An inverse variation involving x and
y is a function in which the product of xy is a
nonzero constant. Another way of writing this is
k xy
k is the constant of variation
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Definition y varies inversely as x means that
y where k is the constant of
variation.
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Examples of Inverse Variation
Note X increases, and Y decreases.
What is the constant of variation of the table
above?
Since y we can say k xy
Therefore (-2)(-18)k or k 36 (72)(0.5)k or k
36 (4)(9)k or k 36 Note k stays constant.
xy 36 or y
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Examples of Inverse Variation
Note X increases, and Y decreases.
What is the constant of variation of the table
above?
Since y we can say k xy
Therefore (4)(16)k or k 64 (-2)(-32)k or k
64 (-2)(-32)k or k 64 Note k stays constant.
xy 64 or y
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Is this an inverse variation? If yes, give the
constant of variation (k) and the equation.
Yes! k -2(-4) or 8 k 4(2) or 8 k 8(1) or
8 k 16(0.5) or 8 Equation? xy 8 or y
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Is this an inverse variation? If yes, give the
constant of variation (k) and the equation.
NO! The constant of variation cannot be 0!
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Is this an inverse variation? If yes, give the
constant of variation (k) and the equation.
Yes! k 2/3(27) or 18 k 2(9) or 18 k -3(-6)
or 18 k 9(2) or 18 Equation? xy 18 or y
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Using Inverse Variation
When x is 2 and y is 4, find an equation that
shows x and y vary inversely.
2 step process
1st Find the constant variation k xy k 2(4)
k 8
2nd Use xy k. xy 8 OR y

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Using Inverse Variation
When x is 3 and y is 12, find an equation that
shows x and y vary inversely.
2 step process
1st Find the constant variation k xy k
3(12) k 36
2nd Use xy k. xy 36 OR y


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Using Inverse Variation to find Unknowns
Given that y varies inversely with x and y -30
when x-3. Find y when x 8. HOW???
2 step process
1. Find the constant variation. k xy or k
-3(-30) k 90
2. Use k xy. Find the unknown (y).
90 xy so 90 8y y 11.25
Therefore x 8 when y11.25
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Using Inverse Variation to find Unknowns
Given that y varies inversely with x and y 20
when x4. Find y when x 10. HOW???
2 step process
1. Find the constant variation. k xy or k
4(20) k 80
2. Use k xy. Find the unknown (y).
80xy so 80 10y y 8
Therefore x 10 when y8
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Using Inverse Variation to solve word problems
Problem The time t that it takes a plane to
reach a certain destination varies inversely as
the average speed s of the plane. It took this
plane 5 hours to reach the given destination when
it traveled at an average speed of 150 mi/hr.
What was the average speed of the plane if it
took 4 hours to reach the same destination?
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Write the equation that relates the variables
then solve.
Problem The time t that it takes a plane to
reach a certain destination varies inversely as
the average speed s of the plane. It took this
plane 5 hours to reach the given destination when
it traveled at an average speed of 150 mi/hr.
What was the average speed of the plane if it
took 4 hours to reach the same destination?
t(time) varies inversely as s(speed) so Time
is the y variable and Speed is the x
variable
K xy K 150(5) K 750 The equation is 750
xy Now substitute 750 x(4) x 187.5
The average speed of the plane to reach the
destination in 4 hours was 187.5 mi/hr.
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Set up a proportion.
Problem The time t that it takes a plane to
reach a certain destination varies inversely as
the average speed s of the plane. It took this
plane 5 hours to reach the given destination when
it traveled at an average speed of 150 mi/hr.
What was the average speed of the plane if it
took 4 hours to reach the same destination?
150(5) 4x 750 4x x 187.5 mi/hr.
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Inverse Variations
The graph make a hyperbola
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What does the graph of xyk look like?
Let k5 and graph.
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Tell if the following graph is a Inverse
Variation or not.
Yes
No
No
No
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Tell if the following graph is a Inverse
Variation or not.
No
Yes
No
Yes