Integrated Topics
Explore Activity Section 1-12
Inverse Variation
On the graph paper provided in this activity, draw as many different rectangles
as you can that have an area of 24 square units.
1.
Write the base and height of each rectangle you draw.
base
height
area
24
24
24
24
24
24
24
24
2.
There are three quantities in this table. Name them.
3.
Which of the quantities are variables? Which is a constant?
4.
Let the letter b represent the length of the base and the letter h represent
the height of the rectangle. Since the area is always 24 we can write:
bh = ______
5.
What happens if you multiply the base in each row of the table by the
corresponding height in that row?
6.
What is the smallest base listed in your table above?_____
What is the largest base listed in your table above?_______
What is the smallest height listed in your table above?_____
What is the largest height listed in your table above?_______
Use the answers to the questions above to create an appropriate viewing window in
your calculator:
WINDOW
Xmin =
Xmax =
Xscl =
Ymin =
Ymax =
Yscl =
Enter the values for the base in list 1 and the height values in list 2 of your
graphing calculator. Use the steps below to create a scatter plot.
STAT
EDIT
Cursor up to L1 hit clear - enter
Cursor to L2 hit clear - enter
Enter the values for the base in L1
Enter the values for the height in L2
hit 2nd y= (STAT PLOT)
Enter
Turn the Statplot ON
cursor to the picture of the scatter plot
enter
Hit GRAPH
Sketch and label the plot on the grid below.
What do you notice about the shape of the scatter plot?
The relationship you are working with in this problem set is called an inverse
variation. An inverse variation is a relationship between two variables that
can be expressed in the form:
[one variable] times [other variable].= [constant]
You should notice that as the value of one variable increases,
the value of the other variable decreases.
An inverse variation has the following properties:
 For every pair of variable values, the product formed when multiplying the
value of one variable times the corresponding value of the other variable is
always the same.
This number is called the constant of variation.
For the example you did at the beginning of this problem set, we will say that
the base of the rectangle (b) varies inversely as the height (h).
This relationship can be represented by the equation bh = 24 , where the
number 24 is called the constant of variation.
In general, if y varies inversely as x, we will write xy = k, where k is the
constant of variation.
Examine the relationship between the variables in the given table:
speed
2
3
4
6
8
walking
mph mph mph mph
mph
(s)
Time to travel 6 hrs 4 hrs 3 hrs 2 hrs 1.5 hrs
some distance
(t)
7.
Is this relationship an inverse variation? Justify your answer.
8.
What number do you get if you multiply the variables in each column?
9.
This number is called the constant of variation. Therefore the constant
of variation is _________
The table below contains the attendance figures for a play when different prices
are charged.
Ticket Price (p)
$2
$3
$4
$6
$9
$12
Attendance (A)
180
120
90
60
40
30
10.
If the relationship between ticket price and attendance is an inverse
variation, the product of the ticket price and attendance must always be
the same. Is it? If so, what is that product?
11.
Write an equation relating the variables. ______________
12.
Input the data from the table into lists in your graphing calculator (use the
steps from problem #5), then make a scatter plot to confirm the fact that
the scatter plot forms the same shape as in problem #5.
A lever is set up to lift a box off the ground as shown in the diagram below.
fulcrum
The force (f) necessary to lift the box off the ground is varies inversely as the
distance (d) you are from the fulcrum.
(The fulcrum is the triangular piece on which the board you are pushing on
rests.)
Suppose that it takes 300 pounds of force to lift the box when you are 2 feet
from the fulcrum.
13.
What is the constant of variation? (Remember in an inverse variation the
product of the variables is a constant.)
14.
Write the equation relating the variables.
15.
If you are at a distance of 3 feet from the fulcrum, how much force
will you need to use?
Integrated Topics
Independent Practice Section 1-12
Direct and Inverse Variation Problems
For each relationship, represented by a table, equation, or graph below, state
whether the relationship between the two variables is a direct variation,
an inverse variation,
or neither.
If the relationship is a direct or inverse variation, give the constant of
variation.
1.
People
attending
Prizes
needed
2.
Items
bought
Total
Cost
3
5
6
8
9
10
6
10
12
16
18
20
2
4
6
8
10
12
$20
$32
$42
$54
$68
$80
20
30
40
45
60
90
18
12
9
8
6
4
3.
Speed
(kph)
Time to
get to
NY (hr)
4.
y = 20x
5.
xy = 100
Solve the following problems.
6.
7.
8.
Suppose that the fine for overdue library books varies directly as the
number of days the book is overdue. Suppose a book that is overdue
for five days has a fine of $0.90.
a.
Determine the constant of variation.
b.
What will the fine be for a book that is overdue for 12 days?
Suppose that the height that a ball bounces varies directly as the
height the ball is dropped from. Suppose that when a ball is dropped
from a height of 10 feet it bounces 8 feet.
a.
What is the constant of variation?
b.
If a ball is dropped from a height of 16 feet, how high will the
ball bounce?
Write an equation for each direct or inverse variation given in the
chart.
a.
x
4
5
6
7
y
10 8 6 2/3 7 1/8
Is the relationship direct or inverse?_______
How do you know?
What is the constant of variation?____________
Write an equation that relates the variables?
b.
x
y
4
15
5
30
6
45
7
60
Is the relationship direct or inverse?_______
How do you know?
What is the constant of variation?____________
Write an equation that relates the variables?