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A variation is a relationship between two variables
Now, go to the class site and
take the mini quiz on direct variation
Now go to the class site and take the
inverse variation mini quiz
Not
Direct Variations
Since y = kx, we can think of a direct variation as
an equation in slope intercept form, with no "b"
y = kx y = mx k = x
So, every direct variation is a straight line with
a y-intercept of (0,0)
The line must pass through the origin
Given that y varies directly with x and y = 6 when x = -5, what is y when x = -8
First, make a chart:
x y
-5 6
-8 y
Now, since we have a direct
variation, we know
6 = k(-5)
-6/5 = k
Plug -8 into our formula now
and we get:
y = -6/5(-8)
y = -48/5
The following table shows a direct variation. Find the value of k
Since this is a direct variation,
we can plug the numbers into
the formula and solve for k:
y = kx
12 = k(6)
2 = k
x y
6 12
7 14
10 20
Direct Variation
Since this value of k works for all of our pairs,
2 is the constant of variation
A direct variation follows the relationship
y = kx
Where k is a constant number, called the
constant of variation
A car uses 8 gallons of gas to drive 200 miles, how much gas will the car use to go 425 miles?
First, make a chart:
Now, since we have a direct
variation, we know
290 = k(8)
25 = k
gas miles
8 200
x 425
When this relationship exists, we say
"y varies directly as x"
Plug 425 into our formula now
and we get:
425 = 25x
17 = x
Does the following table show a direct variation?
In order for the table to be a direct variation,
k must be the same for all of the coordinate pairs
So, the car would use 17 gallons of gas for 425 miles
example: y = 3x
Since k is the same for all
4 pairs, this is a direct variation
and the constant of variation
is 3/2
x y
4 6
8 12
12 18
16 24
(4,6)
6 = k(4)
6/4 = 3/2 = k
(6,12)
12 = k(8)
12/8 = 3/2 = k
(12,18)
18 = k(12)
18/12 = 3/2 = k
(16,24)
24 = k(16)
24/16 = 3/2 = k
Given that y varies inversely with x and y = 2 when x = 8, what is x when y = 4
First, make a chart:
The following table shows a inverse variation. Find the value of k
Now, since we have an inverse
variation, we know
2 = x / 8
16 = x
x y
8 2
x 4
Plug 4 into our formula now
and we get:
4 = 16 / x
4x = 16
x = 4
Direct and Inverse
Since this is a inverse variation,
we can plug the numbers into
the formula and solve for k:
y = k / x
4 = k / 10
40 = k
x y
4 10
5 8
2 20
Since y = k / x, we cannot have 0 for our x value.
Therefore, the graph can never touch the x-axis
Also, the graph never touches the y-axis.
Every inverse variation graph looks similar:
Inverse Variation
Since this value of k works for all of our pairs,
40 is the constant of variation
An inverse variation follows the relationship
y = k/x
Where k is a constant number, called the
constant of variation
When this relationship exists, we say
"y varies inversely as x"
It costs $22.50 per person to rent a bus for 20 passengers. How much would it cost per
person for 36 passengers to rent the bus?
Does the following table show an inverse variation?
example: y = 3/x
In order for the table to be an inverse variation,
k must be the same for all of the coordinate pairs
First, make a chart:
Passengers cost
8 22.5
36 y
Now, since we have an inverse
variation, we know
22.5 = k / 8
180 = k
S0, the bus costs $180 altogether
Since k is the same for all
4 pairs, this is an inverse variation and the constant of variation is 100
x y
4 25
5 20
10 10
50 2
Plug 180 into our formula now
and we get:
y = 180 / 36
y = 5
(10,10)
10 = k / 10
100 = k
(50,2)
2 = k / 50
100 = k
(4,25)
25 = k / 4
100 = k
(5,20)
20 = k / 5
100 = k
So, if 36 people rent the bus, it would cost them each $5 per person
Variation