Transcript Direct Variation

Direct Variation
What is it and how do I know when I see it?
Definition:
(Take notes – write this down)
Y varies directly as x means that y = kx
where k is the constant of proportionality.
y
Another way of writing this is k =
x
In other words:
* As X increases in value, Y increases or
* As X decreases in value, Y decreases.
Your Turn!
What
 What
 What
 What
 What

is Direct Variation?
happens when Y increases?
happens when X decreases?
is the formula?
is the constant of proportionality?
Examples of Direct Variation:
X
6
7
8
Y
12
14
16
Note: X increases,
6,7,8
And Y increases.
12, 14, 16
What is the constant of proportionality of the table above?
y
Since y = kx we can say k 
Therefore:
x
12/6=k or k = 2
14/7=k or k = 2
16/8=k or k =2
Note k stays constant.
y = 2x is the
equation!
Your turn!

Explain how you would find the constant
of proportionality
Examples of Direct Variation:
X (hours)
1
2
3
4
Y (wages)
5
10
15
20
Let’s assume that a girl has
decided to baby sit for $5
per hour.
Make a table of the baby
sitters wages. Use X for the
# of hours and Y for the
wages.
What is the constant of proportionality of the table above?
Since y = kx we can say
Therefore: k  y
x
5/1=k or k = 5
10/2=k or k = 5
15/3=k or k =5
20/4=k or k =5
Note k stays constant.
y = 5x is the
equation!
Examples of Direct Variation:
Note: X decreases,
X
30
15
9
Y
10
5
3
30, 15, 9
And Y decreases.
10, 5, 3
What is the constant of proportionality of the table above?
y
Since y = kx we can say k 
Therefore:
x
10/30=k or k = 1/3 5/15=k or k = 1/3
3/9=k or k =1/3
Note k stays constant.
y = 1/3x is the
equation!
Examples of Direct Variation:
X
40
16
4
Y
10
4
1
Note: X decreases,
40, 16, 4
And Y decreases.
10, 4, 1
What is the constant of proportionality of the table above?
y
k

Since y = kx we can say
Therefore:
x
y = ¼ x is the
10/40 =k or k = ¼
y/x = ¼ or k = ¼
equation!
4/16 =k or k = ¼
Note k stays constant.
What is the constant of proportionality
for the following direct variation?
X
4
8
6
3
3
2
½
4
1.
2.
3.
4.
Y
8
16
12
6
Answer
Now
0%
0%
3
1
2
3
4
5
6
7
8
9
10
11
12
21
22
23
24
25
26
27
28
29
30
31
32
0%
2
1/
2
13
14
15
0%
16
17
4
18
19
20
Is this a direct variation? If yes, give the
constant of proportionality (k) and the equation.
X
4
8
12
18
Y
6
12
18
27
Yes!
k = 6/4 or 3/2
Equation?
y = 3/2 x
Is this a direct variation? If yes, give the
constant of proportionality (k) and the equation.
X
10
6
4
2
Y
25
15
10
5
Yes!
k = 25/10 or 5/2
k = 10/4 or 5/2
Equation?
y = 5/2 x
Is this a direct variation? If yes, give the
constant of proportionality (k) and the equation.
X
15
3
1
2
Y
5
26
75
150
No!
The k values are
different!
Which is the equation that describes the
following table of values?
0%
2
3
4
5
6
7
8
9
10
11
12
21
22
23
24
25
26
27
28
29
30
31
32
0
y
13
14
15
0%
20
=
y
=
y
1
0%
2x
-2
x
0%
=
Answer
Now
xy
4.
x
3.
Y
5
1
6
10
=
2.
X
10
2
12
20
½
y = 3x
y = 2x
y= ½x
xy = 200
1.
16
17
18
19
20
Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 28 when x=7,
Find x when y = 52.
HOW???
2 step process
1. Find the constant variation
k = y/x or k = 28/7 = 4
k=4
X
Y
7
28
?
52
2. Use y = kx. Find the unknown (x).
52= 4x or 52/4 = x
x= 13
Therefore:
X =13 when Y=52
Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 3 when x=9,
Find y when x = 40.5.
HOW???
2 step process
1. Find the constant variation.
k = y/x or k = 3/9 = 1/3
K = 1/3
2. Use y = kx. Find the unknown (x).
y= (1/3)40.5
y= 13.5
X
Y
9
3
40.5
?
Therefore:
X =40.5 when
Y=13.5
Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 6 when x=5,
Find y when x = 8.
HOW???
2 step process
1. Find the constant variation.
k = y/x or k = 6/5 = 1.2
k = 1.2
X
Y
5
6
8
?
2. Use y = kx. Find the unknown (x).
y= 1.2(8)
Therefore:
x= 9.6
X =8 when Y=9.6
Using Direct Variation to solve word problems
Problem:
A car uses 8 gallons of
gasoline to travel 240
miles. How much
gasoline will the car use
to travel 400 miles?
Step Two: Find the constant
variation and equation:
k = y/x or k = 240/8 or 30
y = 30x
Step One: Find points in table
X (gas) Y (miles)
8
240
?
400
Step Three: Use the equation
to find the unknown.
400 =30x
400 =30x
30
30
or x = 13.33
Using Direct Variation to solve word problems
Problem:
Julie wages vary
directly as the number
of hours that he works.
If her wages for 5 hours
are $29.75, how much
will they be for 30 hours
Step Two: Find the constant
variation.
k = y/x or k = 29.75/5 = 5.95
Step One: Find points in table.
X(hours) Y(wages)
5
29.75
30
?
Step Three: Use the equation
to find the unknown. y=kx
y=5.95(30) or Y=178.50
Direct Variation and its graph
With direction variation the equation
is y = kx
Note: k is the constant therefore the graph will always go
through…
the ORIGIN!!!!!
Tell if the following graph is a Direct Variation or not.
No
No
Yes
No
Tell if the following graph is a Direct Variation or not.
No
Yes
Yes
No
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