warm up determine the asymptotes for: lesson 3-8 direct, inverse & joint variation objective: to...
TRANSCRIPT
Warm up
Determine the asymptotes for:
)2(
)3)(2()(
xx
xxxf
Lesson 3-8 Direct, Inverse & Joint Variation
Objective: To recognize and use direct variation to solve problems
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:
* As x increases in value, y increases or
* As x decreases in value, y decreases.
y
x
X Y 30 10 15 5 9 3
Note: X decreases,
30, 15, 9
And Y decreases.
10, 5, 3
What is the constant of variation of the table above?Since y = kx we can say
Therefore:
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!
yk
x
Examples of Direct Variation:
Direct Variation
y1
x1
y2
x2
Direct variation uses the following formula:
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
2.4
15
x
Direct Variation
Example:If y varies directly as the square of x and y = 30 when x = 4, find x when y=270.
y=kx2
30=k42
k=1.875270=1.875x2
x=12 12
270
4
3022
xx
Inverse Variation
Inverse is very similar to direct, but in an inverse relationship as one value goes up, the other goes down.
Inverse Variation
If y varies inversely as x, then
for some constant k.x
ky
k is still called the constant of variation.
Inverse VariationWith Direct variation we
Divide our x’s and y’s. In Inverse variation we will
Multiply them.x1y1 = x2y2
Inverse Variation
If y varies inversely with x and y = 12 when x = 2, find y when x = 8.
x1y1 = x2y2
2(12) = 8y 24 = 8y y = 3
Inverse Variation
If y varies inversely as x and x = 18 when y = 6, find y when x = 8.
18(6) = 8y 108 = 8y
y = 13.5
Practice
If t varies inversely as q. If t = 240 when q = 0.01, then find the value of t when q = 8
Two rectangles have the same area. The length of a rectangle varies inversely as the width. If the length of a rectangle is 20 ft when the width is 8 ft, find the length of the rectangle when the width is 10 ft.?
Joint and Combined Variation
Joint variation is like direct variation but it involves more than one quantity.
Combined variation combines both direct and inverse variation in the same problem.
Joint and Combined Variation
For example: if z varies jointly with x & y, then z=kxy.
Ex: if y varies inversely with the square of x, then y=k/x2.
Ex: if z varies directly with y and inversely with x, then z=ky/x.
Example
y varies jointly as x and w and inversely as the square of z. Find the equation of variation when y = 100, x = 2, w = 4, and z = 20. Then find k.
2z
kxwy
5000k
Example
If y varies jointly as x and z, and y = 12 when x = 9 and z = 3, find z when y = 6 and x = 15.