Entry Task – SHOW WORK
34
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2.2 Direct Variation
Target: I canWrite and interpret direct
variation equations
Definition:
Y varies directly as x means that y = kx where k is the constant of variation.
Another way of writing this is k = In other words:
* the constant of variation (k) in a direct variation is the constant (unchanged) ratio of two variable quantities.
y
x
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
y1
x1
y2
x2
Direct variation uses the following formula:
Direct Variationexample:
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 VariationIf y varies directly as x and y = 10
find x when y =15.
y = 10, x = 2.4 make these y1 and x1
y = 15, and x = ? make these y2 and x2
Direct Variation
if y varies directly as x and y = 10 as x = 2.4, find x when y =15
10
2.4
15
x
Direct Variation
How do we solve this? Cross multiply and set equal.
10
2.4
15
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
2
y
8
Direct Variation
Cross multiply: 96 = 2ySolve for y. 48 = y.
12
2
y
8
X Y 10 30 5 15 3 9
Note: X decreases,
10, 5, 3
And Y decreases.
30, 15, 9
What is the constant of variation of the table above?
Since y = kx we can say 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!
yk
x
Examples of Direct Variation:
Find the value of k.
y=kx Direct variation formula
42 = k(7) Replace y with 15 and x with 5.
42= k(7) 7 7 Divide each side by 56 = k Simplify
Equation?
y = 6x
Y varies directly as x. Find the constant of variation and write an
equation for the direct variation when y = 42 when x = 7
Tell if the following graph is a Direct Variation or not.
No Yes
No No
No Yes
Yes No
Tell if the following graph is a Direct Variation or not.
1. 2.
3.4.
Given that y varies directly with x, and y = 6 when x=-5,
Find y when x = -8. HOW???
2 step processX Y
-5 6
-8 ?
1. Find the constant variation.
k = y/x or k = 6/-5 = -1.2
k = -1.2
2. Use y = kx. Find the unknown (x).
y= -1.2(-8)
x= 9.6
Therefore:
X =-8 when Y=9.6
Using Direct Variation to find unknowns (y = kx)
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 One: Find points in table
X (gas) Y (miles) 8 290 ? 400
Step Two: Find the constant variation and equation:
k = y/x or k = 290/8 or 36.25
y = 36.25 x
Step Three: Use the equation to find the unknown.400 =36.25x 400 =36.25x36.25 36.25or x = 11.03
Homework
P. 71 # 7-47 odds
Challenge - # 54