Download inverse variation

Warm Up
Solve each proportion.
10
1.
2.
3.
4.
2.625
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
Inverse Variation
Objective
Identify, write, and graph inverse operations.
Vocabulary
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.
Inverse variation implies that one quantity will increase
while the other quantity will decrease (the inverse, or
opposite, of increase).
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.
Remember!
A direct variation is an equation that can
be written in the form y = kx, where k is a
nonzero constant.
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.
Example 1A: Identifying an Inverse Variation
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.
1(30) = 30, 2(15) = 30, 3(10) = 30
The product xy is constant, so the relationship is
an inverse variation.
Example 1B: Identifying an Inverse Variation
Tell whether each 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.
Example 1C: Identifying an Inverse Variation
Tell whether each 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.
Try 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.
Try 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.
Try 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.
Helpful Hint
Since k is a nonzero constant, xy ≠ 0.
Therefore, neither x nor y can equal 0, and no
solution points will be on the x- or y-axes.
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). This is
because (0, 0) can never be a
solution of an inverse
variation equation.
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
Write the rule for constant of variation.
= –12(0.5)
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.
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
1
6 undef. –6
2
4
–3 –1.5
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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Try 2
Write and graph the inverse variation in which y = when
x = 10.
Step 1 Find k.
k = xy
= 10
Write the rule for constant of variation.
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.
Try 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
Try 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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Remember!
Recall that sometimes domain and range are
restricted in real-world situations.
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. Use the graph to estimate
the volume of the gas when the pressure is 40
atmospheric units.
Step 1 Solve the function for y so you can graph it.
xy = 100
Divide both sides by x.
Example 3 Continued
Step 2 Decide on a reasonable domain and range.
Pressure is never negative and x ≠ 0
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
y
10
10
20
5
30
3.34
40
2.5
Example 3 Continued
Step 4 Plot the points. Connect them with a smooth curve.
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Step 5 Find the y-value where x = 40. When the pressure is 40
atm, the volume of gas is about 2.5 mm3.
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.
Example 4: Using the Product Rule
Let
as x. Find
and
Let y vary inversely
Write the Product Rule for Inverse Variation.
Simplify.
Solve for
Simplify.
by dividing both sides by 5.
Try 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.