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10-1 Inverse Variation
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Warm Up
California Standards
Lesson Presentation
10-1 Inverse Variation
Warm Up
Solve each proportion.
1.
10
2.
3.
2.625
4.
4.2
2.5
5. The value of y varies directly with x, and y = –6
when x = 3. Find y when x = –4. 8
6. The value of y varies directly with x, and y = 6
when x = 30. Find y when x = 45. 9
10-1 Inverse Variation
California
Standards
Preparation for
13.0
Students add, subtract, multiply, and divide
rational expressions and functions.
Students solve both computationally and
conceptually challenging problems by using
these techniques.
Also covered: 17.0
10-1 Inverse Variation
Vocabulary
inverse variation
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 =
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.
form.
10-1 Inverse Variation
Additional Example 1C: Identifying an Inverse
Variation
Tell whether the relationship is an inverse
variation. Explain.
2xy = 28
xy = 14
Find xy. Since xy is multiplied by 2,
divide both sides by 2 to undo the
multiplication.
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 =
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.
form.
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 intercept the x- or y-axes.
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
y
1.5
–2 –1
3
0
1
6 undef. –6
2
4
–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 = when x = 10.
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 mi/h 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
Remember!
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 5 for
3 for
and 10 for
Simplify.
Solve for
Simplify.
by dividing both sides by 5.
.
10-1 Inverse Variation
Check It Out! Example 4
Let
and
inversely as x. Find
Let y vary
Write the Product Rule for Inverse
Variation.
Substitute 2 for
–4 for
and –6 for
Simplify.
Solve for
Simplify.
by dividing both sides by –4.
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
for
, 60 for
and 4.3
Simplify.
Solve for
Simplify.
The child weighs 80.625 lb.
by dividing both sides by 3.2.
10-1 Inverse Variation
Lesson Quiz: Part I
1. Write and graph the inverse variation in which
y = 0.25 when x = 12.
10-1 Inverse Variation
Lesson Quiz: Part II
2. The inverse variation xy = 210 relates the length
y in cm to the width x in cm of a rectangle with
an area of 210 cm2. Determine a reasonable
domain and range and then graph this inverse
variation.
10-1 Inverse Variation
Lesson Quiz: Part III
3. Let x1= 12, y1 = –4, and y2 = 6, and let y vary
inversely as x. Find x2.
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