1. What is it and how do I know when I see it?
Inverse Variation
What is it and how do I know when I see it?

2. 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.

3. 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

4. Definition:
y varies inversely as x means that
y =
where k is the constant of variation.

5. 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 =

6. 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 =

7. Is this an inverse variation
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 =

8. The constant of variation cannot be 0!
Is this an inverse variation? If yes, give the constant of variation (k) and the equation.
NO!
The constant of variation cannot be 0!

9. Is this an inverse variation
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 =

10. 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 =

11. 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 =

12. Using Inverse Variation to find Unknowns
Given that y varies inversely with x and y = -30 when x= Find y when x = 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

13. Using Inverse Variation to find Unknowns
Given that y varies inversely with x and y = 20 when x=4. Find y when x = 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

14. Using Inverse Variation to solve word problems
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?

15. 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 mi/hr.

16. 150(5) = 4x 750 = 4x x = 187.5 mi/hr. 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 = mi/hr.

17. Inverse Variations
The graph make a hyperbola

18. What does the graph of xy=k look like?
Let k=5 and graph.

19. Tell if the following graph is a Inverse Variation or not.
Yes No No No

20. Tell if the following graph is a Inverse Variation or not.
Yes No
Yes