Inverse Variation What is it and how do I know when I see it?
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Transcript of Inverse Variation What is it and how do I know when I see it?
Inverse VariationWhat is it and how do I know when I see it?
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.
Inverse Variation
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
Definition:
y varies inversely as x means that
y =
where k is the constant of variation.
€
k
x
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 = €
k
x
€
36
x
Note:
X increases,
and Y decreases.
What is the constant of variation of the table above?
Examples of Inverse Variation:
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 =
€
k
x
€
64
x
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 =
Is this an inverse variation? If yes, give the constant of variation (k) and the equation.
€
8
x
NO!
The constant of variation cannot be 0!
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 =
Is this an inverse variation? If yes, give the constant of variation (k) and the equation.
€
18
x
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 =
€
8
x
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 =
€
36
x
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 y=11.25
Using Inverse Variation to find Unknowns
Given that y varies inversely with x and y = 20 when x=4. 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).
80=xy so 80= 10y
y= 8
Therefore:
x = 10 when y=8
Using Inverse Variation to find Unknowns
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?
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?
Write the equation that relates the variables then solve.
t(time) varies inversely as s(speed)so
Time is the y variableand
Speed is the x variable
K = xyK = 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.
Set up a proportion.
x1
y2
=x2
y1
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
4=
x
5150(5) = 4x750 = 4xx = 187.5 mi/hr.
Inverse Variations
The graph make a hyperbola
What does the graph of xy=k look like?
Let k=5 and graph.
6
4
2
-2
-4
-6
-10 -5 5 10
f x( ) = 5
x
Tell if the following graph is a Inverse Variation or not.
No Yes
No No
Yes No
Yes No
Tell if the following graph is a Inverse Variation or not.