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10-1 Inverse Variation
A relationship that can be written in the form y = , where k is a
nonzero constant and x ≠ 0, is an inverse variation. The constant k
is the constant of variation.
Multiplying both sides of y = by x gives xy = k. So, for any inverse
variation, the product of x and y is a nonzero constant.
10-1 Inverse Variation
There are two methods to determine whether a relationship
between data is an inverse variation. You can write a function
rule in y = form, or you can check whether xy is a constant for
each ordered pair.
10-1 Inverse Variation
Additional Example 1A: Identifying an Inverse Variation
Tell whether the relationship is an inverse variation. Explain.
Method 1: Write a function rule.
Can write in y =
form.
The relationship is an inverse variation.
Method 2: Find xy for each ordered pair.
1(30) = 30, 2(15) = 30, 3(10) = 30
The product xy is constant, so the relationship is an inverse
variation.
10-1 Inverse Variation
Additional Example 1B: Identifying an Inverse Variation
Tell whether the relationship is an inverse variation. Explain.
Method 1: Write a function rule.
y = 5x
Cannot write in y =
form.
The relationship is not an inverse
variation.
Method 2: Find xy for each ordered pair.
1(5) = 5, 2(10) = 20, 4(20) = 80
The product xy is not constant, so the relationship is not an
inverse variation.
10-1 Inverse Variation
Additional Example 1C: Identifying an Inverse Variation
Tell whether the relationship is an inverse variation. Explain.
2xy = 28
Find xy. Since xy is multiplied by 2, divide
both sides by 2 to undo the multiplication.
xy = 14
Simplify.
xy equals the constant 14, so the relationship is an inverse
variation. The equation can be written in the form y =
10-1 Inverse Variation
Check It Out! Example 1A
Tell whether each relationship is an inverse variation. Explain.
Method 1: Write a function rule.
y = –2x
Cannot write in y =
form.
The relationship is not an inverse
variation.
Method 2: Find xy for each ordered pair.
–12(24) = –228, 1(–2) = –2, 8(–16) = –128
The product xy is not constant, so the relationship is not an
inverse variation.
10-1 Inverse Variation
Check It Out! Example 1B
Tell whether each relationship is an inverse variation. Explain.
Method 1: Write a function rule.
Can write in y =
form.
The relationship is an inverse variation.
Method 2: Find xy for each ordered pair.
3(3) = 9, 9(1) = 9, 18(0.5) = 9
The product xy is constant, so the relationship is an inverse
variation.
10-1 Inverse Variation
Check It Out! Example 1C
Tell whether each relationship is an inverse variation. Explain.
2x + y = 10
Cannot write in y =
form.
The relationship is not an inverse variation.
10-1 Inverse Variation
Helpful Hint:
Since k is a nonzero constant, xy ≠ 0.
Therefore, neither x nor y can equal 0, and the graph will not
touch the x-axis or y-axis.
10-1 Inverse Variation
An inverse variation can also be
identified by its graph. Some
inverse variation graphs are
shown. Notice that each graph
has two parts that are not
connected.
Also notice that none of the
graphs contain (0, 0). In other
words, (0, 0) can never be a
solution of an inverse variation
equation.
10-1 Inverse Variation
Additional Example 2: Graphing an Inverse Variation
Write and graph the inverse variation in which y = 0.5 when
x = –12.
Step 1 Find k.
k = xy
= –12(0.5)
Write the rule for constant of variation.
Substitute –12 for x and 0.5 for y.
= –6
Step 2 Use the value of k to write an inverse variation equation.
Write the rule for inverse variation.
Substitute –6 for k.
10-1 Inverse Variation
Additional Example 2 Continued
Write and graph the inverse variation in which y = 0.5 when
x = –12.
Step 3 Use the equation to make a table of values.
x
–4
–2
y
1.5
3
–1
0
6 undef.
1
2
4
–6
–3
–1.5
10-1 Inverse Variation
Additional Example 2 Continued
Write and graph the inverse variation in which y = 0.5 when
x = –12.
Step 4 Plot the points and connect them with smooth curves.
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10-1 Inverse Variation
Check It Out! Example 2
Write and graph the inverse variation in which y = when
x = 10.
Step 1 Find k.
k = xy
Write the rule for constant of variation.
= 10
Substitute 10 for x and
for y.
= 5
Step 2 Use the value of k to write an inverse variation equation.
Write the rule for inverse variation.
Substitute 5 for k.
10-1 Inverse Variation
Check It Out! Example 2 Continued
Write and graph the inverse variation in which y = when
x = 10.
Step 3 Use the equation to make a table of values.
x
y
–4
–2
–1
0
1
2
4
–1.25
–2.5
–5
undef.
5
2.5
1.25
10-1 Inverse Variation
Check It Out! Example 2 Continued
Write and graph the inverse variation in which y =
x = 10.
when
Step 4 Plot the points and connect them with smooth curves.
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10-1 Inverse Variation
Additional Example 3: Transportation Application
The inverse variation xy = 350 relates the constant speed x in
miles/hour to the time y in hours that it takes to travel 350
miles. Determine a reasonable domain and range and then
graph this inverse variation.
Step 1 Solve the function for y.
xy = 350
Divide both sides by x.
10-1 Inverse Variation
Additional Example 3 Continued
Step 2 Decide on a reasonable domain and range.
x>0
Speed is never negative and x ≠ 0.
y>0
Because x and xy are both positive, y is also
positive.
Step 3 Use values of the domain to generate reasonable
ordered pairs.
x
20
40
60
80
y
17.5
8.75
5.83
4.38
10-1 Inverse Variation
Additional Example 3 Continued
Step 4 Plot the points. Connect them with a smooth curve.
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10-1 Inverse Variation
Recall that sometimes domain and range are restricted in
real-world situations.
10-1 Inverse Variation
Check It Out! Example 3
The inverse variation xy = 100 represents the relationship
between the pressure x in atmospheres (atm) and the volume y
in mm3 of a certain gas. Determine a reasonable domain and
range and then graph this inverse variation.
Step 1 Solve the function for y.
xy = 100
Divide both sides by x.
10-1 Inverse Variation
Check It Out! Example 3 Continued
Step 2 Decide on a reasonable domain and range.
x>0
Pressure is never negative and x ≠ 0
y>0
Because x and xy are both positive, y is also
positive.
Step 3 Use values of the domain to generate reasonable pairs.
x
10
20
30
40
y
10
5
3.34
2.5
10-1 Inverse Variation
Check It Out! Example 3 Continued
Step 4 Plot the points. Connect them with a smooth curve.
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10-1 Inverse Variation
The fact that xy = k is the same for every ordered pair in any
inverse variation can help you find missing values in the
relationship.
10-1 Inverse Variation
10-1 Inverse Variation
Additional Example 4: Using the Product Rule
Let
as x. Find
and
Let y vary inversely
Write the Product Rule for Inverse Variation.
Substitute .
Simplify.
Solve by dividing both sides by 5.
Simplify.
10-1 Inverse Variation
Check It Out! Example 4
Let
Find
and
Let y vary inversely as x.
Write the Product Rule for Inverse Variation.
Substitute.
Simplify.
Solve by dividing both sides by –4.
Simplify.
10-1 Inverse Variation
Additional Example 5: Physical Science Application
Boyle’s law states that the pressure of a quantity of gas x varies
inversely as the volume of the gas y. The volume of gas inside a
container is 400 in3 and the pressure is 25 psi. What is the pressure
when the volume is compressed to 125 in3?
Use the Product Rule for Inverse
Variation.
(400)(25) = (125)y2
Substitute 400 for x1, 125 for x2,
and 25 for y1.
Simplify.
Solve for y2 by dividing both sides
by 125.
10-1 Inverse Variation
Additional Example 5 Continued
Boyle’s law states that the pressure of a quantity of gas x varies
inversely as the volume of the gas y. The volume of gas inside a
container is 400 in3 and the pressure is 25 psi. What is the pressure
when the volume is compressed to 125 in3?
When the gas is compressed to 125 in3, the pressure
increases to 80 psi.
10-1 Inverse Variation
Check It Out! Example 5
On a balanced lever, weight varies inversely as the distance from
the fulcrum to the weight. The diagram shows a balanced lever.
How much does the child weigh?
10-1 Inverse Variation
Check It Out! Example 5 Continued
Use the Product Rule for Inverse Variation.
Substitute 3.2 for
, ,60 for
and 4.3 for
Simplify.
Solve for
Simplify.
The child weighs 80.625 lb.
by dividing both sides by 3.2.