Transcript The Product Rule for inverse variations

Math 8H
11-1
Inverse Variation
Algebra 1
Glencoe McGraw-Hill
JoAnn Evans
You learned in chapter 4 that a direct variation
equation represents a constant rate of change.
The direct variation equation
y  kx
states that as y increases, x increases.
We say that y varies directly as x.
You learned that every direct variation equation will
graph as a straight line that passes through the origin.
(0, 0)
Direct variation equations are linear equations.
x
Today you’ll learn about inverse variation
equations. The relationship between x and y
is different in an inverse variation.
Two quantities that vary inversely always
have the same product.
xy  k
The product, k, is called the constant of variation.
The constant of variation cannot be equal to zero.
If the two variables x and y always have the same
product, this will be true:
as y increases, x decreases.
xy k
1  36  36
2  18  36
3  12  36
4  9  36
6  6  36
8  4.5  36
9  4  36
10  3.6  36
12  3  36
16  2.25  36
18  2  36
36  1  36
xy  k
Let’s say that k, the constant of
variation, is 36.
The two quantities x and y would
always have a product of 36.
As you look down the column
of products, as x increases,
y decreases in order for the
product to remain the same.
xy  36
is the inverse variation equation
to describe this relationship.
Graph the inverse variation equation xy = 36.
2 18
3 12
4 9
6 6
9 4
12 3
• ••
•
•
•
•
-2
-4
-6
2 4 6
• •
y
-2
-3
-4
-6
-9
-12
-18
•
•
•
•
•
x
-18
-12
-9
-6
-4
-3
-2
The graph of an inverse variation isn’t a straight line
like the graph of a direct variation. Instead, it’s a
sweeping curve that will approach the x- and y- axes,
but never touch them.
Grandma’s house is 100 miles away. The speed you use to drive
there will vary inversely with the time it takes to get there.
In other words, as the speed increases, the time decreases.
The equation rt = 100
can be used to represent
you driving there. At 50
miles per hour, the trip
will take 2 hours.
rt = k
50 ∙ 2 = 100
At 40 mph, how long
will the trip take?
40 ∙ t = 100
At 20 mph, how long
will the trip take?
2.5 hrs
At 80 mph, how long
will the trip take?
80 ∙ t = 100
speed and time
vary inversely, but
their product (the
distance) is always
100
20 ∙ t = 100
5 hrs
At 10 mph, how long
will the trip take?
1.25 hrs
10 ∙ t = 100
10 hours
Your rate of speed can’t
be negative, so it’s only
logical to use positive
values for r.
10
5
5 10
•
Why is this graph only a
single curve, unlike the
previous graph?
•
80 1.25
•
•
•
t
50
20
10
5
50 2
•
r
2
5
10
20
t
r
Graph the inverse variation if y varies inversely as x.
y = 4 when x = 5.
Inverse variation equation: xy = k
(5)(4) = k
20 = k
x
-10
-5
-4
-2
2
4
5
10
y
-2
-4
-5
-10
10
5
4
2
xy = 20
The Product Rule for inverse variations:
If (x1, y1) and (x2, y2) are solutions to an
inverse variation, then x1y1 = k and x2y2 = k.
If the products of x1y1 and x2y2 are both
equal to the same constant, then they
would also be equal to each other.
x1y1  x2y2
Write an inverse variation equation that relates x and y.
Assume that y varies inversely as x. Then solve.
If y = 12.5 when x = 4, find x when y = 10.
x1y1  x2y2
 412.5
 x 10
50  10x
Product Rule for inverse variations
Substitute known values.
Divide each side by 10.
5x
x is 5 when y is 10
If y varies inversely as x and y = -7.5 when x = 25,
find y when x = 5.
y = -7.5 when x = 25
x1y1  x2y2
25 7.5
  5 y
187.5  5y
What is y when x = 5?
Product Rule for inverse variations
Substitute known values.
Divide each side by 10.
37.5  y
y is -37.5 when x is 5
If y varies inversely as x and y = 12 when x = 4,
find x when y = 18.
y = 12 when x = 4
x1y1  x2y2
 412
 x 18
48  18x
8
x
3
What is x when y = 18?
Product Rule for inverse variations
Substitute known values.
Divide each side by 18.
8
x is
when y is 18
3
The Malahowski family can drive from Lisle, Illinois, to
Oshkosh, Wisconsin in 4 hours if they drive an average
of 45 miles per hour. How long would it take them if
they increased their average speed to 50 mph?
x1y1  x2y2
 4 45  x 50
180  50x
3.6  x
Trip time would decrease to
3.6 hours if the average speed
was increased to 50 mph.
The manager of Stringer’s Hardware scheduled 8
employees to start and finish taking inventory in an 8
hour period. Assume, like the manager, that all
employees work at the same rate.
Unfortunately, 3 employees called in sick that day. How long
will it take 5 employees to complete the inventory?
x1y1  x2y2
88  5 y
64  5y
12.8  y
The 5 employees will need 12.8
hours for the inventory.
The sound produced by a string inside a piano depends
on the length of the string. The frequency of a
vibrating string varies inversely as its length.
frequency  length = k
If a 2 ft string vibrates 300 cycles per second, what
would be the frequency of a string that is 5 ft long?
x1y1  x2y2
3002  x 5
600  5y
120  y
A 5 ft string will vibrate 120
cycles per second.