Writing Equations of Lines
Chapter 2
Section 4
Day 2
Direct Variation
What is it and how do I know when I see it?
Definition:
Two variables x and y show direct variation
provided f(x) = kx where k is the constant of
variation and k ≠ 0.
y is said to vary directly with x
(see any similarities to f(x) = mx + b?)
Another way of writing this is k =
y
x
In other words:
* As X increases in value, Y increases or
* As X decreases in value, Y decreases.
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 variation 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.
f(x) = 2x 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 variation 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
f(x) = 1/3x is
the equation!
Note k stays constant.
Examples of Direct Variation:
Note: X decreases,
X
-4
-16
-40
Y
-1
-4
-10
-4, -16, -40
And Y decreases.
-1,-4,-10
What is the constant of variation of the table above?
y
Since y = kx we can say k =
Therefore:
x
f(x) = ¼ x is
-1/-4=k or k = ¼
-4/-16=k or k = ¼
the equation!
-10/-40=k or k = ¼ Note k stays constant.
What is the constant of variation for the
following direct variation?
1.
2.
3.
4.
X
4
8
-6
3
2
-2
-½
½
Answer
Now
Y
-8
-16
12
-6
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
X
4
8
12
18
Y
6
12
18
27
Yes!
k = 6/4 or 3/2
Equation?
f(x) = 3/2 x
Is this a direct variation? If yes, give the
constant of variation (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?
f(x) = 5/2 x
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
X
15
3
1
2
Y
5
26
75
150
No!
The k values are
different!
Which of the following is a direct variation?
1.
2.
3.
4.
A
B
C
D
Answer
Now
Which is the equation that describes the
following table of values?
1.
2.
3.
4.
f(x) = -2x
f(x) = 2x
f(x) = ½ x
xy = 200
Answer
Now
X
10
2
12
20
Y
5
1
6
10
Using Direct Variation to find unknowns
(f(x) = 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
2. Use y = kx. Find the unknown (x).
X
Y
7
28
?
52
52= 4x or 52/4 = x
Therefore:
x= 13
x =13 when y=52
Using Direct Variation to find unknowns
(f(x) = 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.
X
Y
k = y/x or k = 6/-5
-5
6
k = -6/5
-8
?
2. Use y = kx. Find the unknown (x).
y= -6/5(-8)
Therefore:
x= 48/5
x =-8 when y = 48/5
Direct Variation and its graph
f(x) = mx +b,
m = slope and b = y-intercept
With direction variation the equation
is f(x) = kx
Note: m = k or the constant and b = 0 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
If you are looking at a graph you can tell whether
the relationship represented on the graph is
directly related if the graph is a straight line that
passes through the origin.
If you are looking at a table you can tell whether
the relationship represented in the table is
directly related if when you divide each y value
by each x value and you get the same number.
If you are looking at an equation you can tell
whether the relationship is represented by the
equation is directly related if it is in the form of
f(x) = kx