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Lesson 64
Identifying, writing, and
graphing inverse variation
Review direct variation

As
y = kx
one variable increases,
the other increases at a
constant rate, k
Inverse variation

Inverse variation is a relationship between 2
variables whose product is a constant.
The equation xy = k or y = k/x, where k
is a nonzero constant, defines an inverse
variation between x and y
 As x increases, y decreases and as x
decreases, y increases
 Look for the words "varies inversely" or "is
inversely proportional to"

Example of inverse variation
 The
relationship between the length
and width of a rectangle with a
constant area is an inverse variation
because in order for the area to stay
the same, if the length increases, the
width must decrease.
 L x W = Area
exploration
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1) draw a rectangle that has a width of 1 unit and a
length of 16 units on graph paper
2)draw different rectangles with the same area but
different widths and lengths
3) make a table and complete it after drawing 5 other
rectangles with the same area
4) what happens to the length of each rectangle as the
width increases?
5) what will the product of the width and length always
be?
6)write an equation solved for y showing this
relationship.
 In
a direct variation, y is equal to
the product of a constant k and x
or y = kx
 In an inverse variation, y is equal
to the quotient of k and x or
 y = k/x
Identifying an inverse variation
Tell whether each relationship is an inverse
variation:
 y/6 = x solve for y
 6(y/6) = (x)6
 y = 6x this a direct variation
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xy = 5
y = 5/x inverse variation
Direct or inverse variation?

x = 36/y

y/12 = x

4y=x

3xy=9
Product rule for inverse
variation
If
(x1, y1) and (x2,y2) are
solutions of an inverse
variation, then x1y1= x2y2
Using the product rule

If y varies inversely as x and y = 3 when x
= 12, find x when y = 9
Use the product rule x1y1= x2y2

(12)(3) = x2(9)

36 = 9x2

4 = x2
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practice
If
y varies inversely as x
and y = 5 when x = 12,
find x when y = 3
Graphing an inverse variation

Write an inverse variation relating x and y
when y = 8 and x = 3. Then graph it.
Find k
 xy = k
 (3)(8) = 24
 k = 24
so xy=24 y = 24/x
 Make a table of values and plot the points

practice
Write
an inverse variation
relating x and y when x = 2
and y = 6. Then graph the
relationship.
Investigation 7
Comparing
direct and
inverse variation
compare
Direct variation is a relationship between 2
variables whose ratio is constant. The
equation y = kx, where k is a nonzero
constant called the constant of variation,
shows direct variation between variables x
and y.
 Identify the constant of variation, given
that y varies directly with x
 y is 10 when x = 2
 y is 3 when x = 6

Word problem
Alex walks 3 miles per hour. If he walks at that
rate for twice as long, he will travel twice as far.
The ratio of the distance and time is always the
same.
 Identify the constant of variation- rate, k = ?
 Write an equation of direct variation that relates
Alex's time to his distance traveled

Inverse variation
In inverse variation, as x increases, y
decreases. An inverse variation describes a
relationship between 2 variables whose
product is a constant. The equation xy=k ,
where k is a nonzero constant, defines an
inverse variation between x and y
 Identify the constant of variation, given
that y varies inversely with x. then write the
constant of variation.
 y is 1 when x is 3
 y is 4 , when x is 1/2

Inverse variation
The
equation of
inverse variation can
be written as y = k/x
or xy = k
Word problem
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Alex lives 4 miles from school. If he walks at a
slower rate than normal, it will take him longer to
reach his destination. In other words, the more
time he spends walking home, the slower he is
actually walking. This represents an inverse
variation.
Identify the constant of variation.
Write an equation of inverse variation that
relates Alex's time to his rate of speed.
graphing
 A direct
 An
variation graph is linear
inverse variation graph is not
linear and never intersects the xaxis