Transcript Alg 2.2 2.2 direct variation

Entry Task – SHOW WORK
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2.2 Direct Variation
Learning Target: I can
Write and interpret direct
variation equations
Definition:
Y varies directly as x means that y = kx
where k is the constant of variation.
y
Another way of writing this is k =
x
In other words:
* the constant of variation (k) in a direct
variation is the constant (unchanged) ratio of two
variable quantities.
When we talk about a direct variation, we are talking
about a relationship where as x increases,
y increases or decreases at a CONSTANT RATE.
Direct Variation
Direct variation uses the
following formula:
y1 y 2

x1 x 2
Direct Variation
example:
if y varies directly as x and y = 10 as x =2.4,
find x when y =15. what x and y go together?
Direct Variation
if y varies directly as x and y = 10
as x = 2.4, find x when y =15
10 15

2.4
x
Direct Variation
How do we solve this? Cross
multiply and set equal.
10 15

2.4
x
Direct Variation
We get: 10x = 36
Solve for x by diving both sides by 10.
We get x = 3.6
Direct Variation
Let’s do another.
If y varies directly with x and y = 12
when x = 2, find y when x = 8.
Set up your equation.
Direct Variation
If y varies directly with x and y = 12
when x = 2, find y when x = 8.
12 y

2 8
Direct Variation
12 y

2 8
Cross multiply: 96 = 2y
Solve for y.
48 = y.
Examples of Direct Variation:
X
10
5
3
Y
30
15
9
Note: X decreases,
10, 5, 3
And Y decreases.
30, 15, 9
What is the constant of variation of the table above?
y
Since y = kx we can say k  x Therefore:
30/10=k or k = 3
15/5=k or k = 3
9/3=k or k =3
Note k stays constant.
y = 3x is the
equation!
Y varies directly as x. Find the constant of
variation and write an equation for the
direct variation when y = 42 when x = 7
Find the value of k.
y=kx
Direct variation formula
42 = k(7)
Replace y with 42 and x with 7.
42= k(7)
7
7
6=k
Divide each side by 7
Simplify
Equation?
y = 6x
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.
1.
2.
No
3.
Yes
4.
Yes
No
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 290
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 = 290/8 or 36.25
y = 36.25 x
Step One: Find points in table
X (gas) Y (miles)
8
290
?
400
Step Three: Use the equation
to find the unknown.
400 =36.25x
400 =36.25x
36.25 36.25
or x = 11.03
Homework
P. 71 #25,26,30-33,35,43-46,51,52
Challenge - # 54