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.