september 6 direct variation. direct variation x is directly proportional to y x varies directly as...
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September 6Direct Variation
DIRECT VARIATIONx is directly proportional to y
x varies directly as y
YYXX
Direct Variation• y = kx• k is the constant of
variation• the graph must go
through the origin (0,0) and must be linear!!
• Therefore it must be in y = kx form. The y-intercept is 0
Direct Variation
Example NonExample
y = 3x
y = .5x-1
y = 2/3x
y = 5
y = 4 – 6x
y = 11x
y = 8.7x
Direct VariationEx 1)If x varies directly as y and
x = 12 when y = 3, write an equation that relates x and y.
Start with: y = kx
Fill in x and y: 3 = k(12)
Solve for k: 3 1
12 4k
Re-write equation with the k value: y = ¼ x
Same problem, new ?Ex 1)If x varies directly as y and x = 12 when y = 3, find x when y
= 10.
y = ¼ xFill in NEW x and y: 10 = ¼ (x)
Solve for x: x = 40
Another way to do the last ?
2
2
1
1
y
x
y
x FIRST: what you are
comparing
NEXT: substitute your values correctly
LAST: cross multiply to solve for missing variable.
12
3 10
x
12(10) 3
40
x
x
2) If y varies directly as x, and y = 28 when x = 7, find x when y = 52
write an equation that relates x and y.
28 52:
7
y
x x x = 13
What is the constant of variation? 4The constant of variation is the reduced fraction.
y = 4x
3) If y varies directly as the square of x, and y = 4 when x = 3, find y
when x = 6Use a proportion…..
2 2 2
4:3 6
y y
x y = 16
write an equation that relates x and y.4
9y x
4) A car uses 8 gallons of gasoline to travel 290 miles.
How much gasoline will the car
use to travel 400 miles?
8
290 400
gas
m les
x
i
11.034 gallons
5) In scuba diving the time (t) it takes a diver to ascend safely to the surface varies directly with the depth (d) of the dive. It takes a minimum of 3 minutes from a safe ascent from 12 feet. Write an equation that relates depth (d) and time (t). Then determine the minimum time for a safe ascent from 1000 feet?
3 1
12 4
t
d 4t d 4 1000
250min
t
t
6) z varies directly with x and y.
z = kxyWrite the equation relating x, y and z if x = 2, y = -6 and z = 24.
24 (2)( 6)k
24 12k
2k
2z xy
8.2 Inverse Variation
INVERSE or indirect VARIATIONy is inversely proportional to x
y varies inversely as x
K is the constant of variation or constant of proportionality
ky
x
XX YY
Inverse Variation
• This is a NON-LINEAR
function (it doesn’t look like y=mx+b)
• It doesn’t even get close to (0, 0)
• k is still the constant of variation
Inverse Variation
When you buy a car, as time (t) increases, the value (v) decreases.
tt vv
The constant of variation, k is the amount that it decreases.
t is the age of the car.
v is the value of the car.
t
kv
Write the model that represents this situation.
6) If y varies inversely as x and when y = 12, x = 10.
x
ky 12
10
k 120k
120y
x
7)The intensity of a light “I” received from a source varies inversely with the distance “d” from the source. If the light intensity is 10 ft-candles at 21 feet, what is the light intensity at 12 feet? Write your equation first.
kId
1021
k 2100k
2100
12I 175I ft candles
Work with partners on the WS
HW: finish WS 5
show all work!