Inverse or Direct Variation Basics
x - 1, 5, 10, 20
y - 5, 25, 50, 100
1) Solve for k
2) Write the function that models that data
3) Is it direct or inverse?
k = 5
y = 5x
direct
If x varies directly as y, and x = 27 when y = 6, find x when y = 2.
Write the equation for x:
Solve for k:
Solve for x:
x = ky
k= 2/9
x = 9
(2a2b) / (b2c) * (b) / (a)
combine numerators and denominators:
simplify:
which letters can't equal 0?
combine: 2a2b2 / ab2c
simplify: 2a / c
a, b, c can't equal 0
((x2 - 2x) / 6) / ((3x - 6) / x )
flip:
factor and combine numerators and denominators:
simplify:
restrictions?
flip: (x(x - 2) / 6) * (x / (3x - 6))
factor and combine: (x2(x - 2)) / (18(x - 2))
simplify: x2 / 18
x can't equal 2
12x6 / 42x4
simplify:
x can't equal what?
is that a hole or an asymptote
simplify: 2x2 / 7
x can't equal 0
0 is a hole
x - 1, 2, 4, 8
y - 40, 20, 10, 5
1) Solve for k
2) Write the function that models that data
3) Is it direct or inverse?
k = 40
y=40 / x
inverse
If y varies inversely as x, and y = 14 when x = 8, find y when x = 7.
Write the equation for y:
Solve for k:
Solve for x:
y = k/x
k = 112
x = 16
((y2 - 2y - 15) / 4) * (8 / (y + 3))
factor and combine numerators and denominators:
simplify:
y-restrictions
factor and combine: (8(y - 5)(y + 3)) / ((4)(y + 3))
simplify: 2(y-5)
y can't equal -3
(7x2 / 12x) / (14x3 / 48y3)
flip:
simplify:
x-restrictions?
flip: (7x2 / 12x) * (48y3 / 14x3)
simplify: 2y3 / x2
x can't equal 0
(x2 + 8x + 15) / (3x + 9)
factor numerator and denominator:
simplify:
x can't equal what number(s):
is that a hole or asymptote?
factor and combine: (x+5)(x+3) / 3(x+3)
simplify: (x + 5) / 3
x can't equal -3
-3 is an asymptote
x - 2, 5, 7, 10
y - 5, 10, 30, 55
Is this a direct or inverse relationship?
neither
y varies jointly with a and b squared. If y = 40 when a = 5 and b = 2:
Write the equation for y:
What is the value of k?
What does y equal if a = 8 and b = 3
y = kab2
k = 2
y = 144
((5n + 15) / (4n + 8)) * ((2n + 4) / (3n + 9))
factor and combine numerators and denominators
simplify:
n-restrictions:
factor and combine: (10)(n + 3)(n + 2)) / ((12(n + 2)(n + 3))
simplify: 5 / 6
n can't equal -2, -3
((x2 + 5x - 24) / (2x + 2)) / ((3x + 24) / (x2 - 8x - 9))
flip:
factor and combine numerators and denominators:
simplify:
x-restrictions?
flip: ((x2 + 5x - 24) / (2x + 2)) * ((x2 - 8x - 9) / (3x + 24))
factor and combine: ((x + 8)(x - 3)(x - 9)(x + 1)) / (6(x + 1) (x + 8))
simplify: ((x - 3)(x - 9)) / 6
x can't equal -1 or -8
((x + 3) / (5x + 20)) * ((x2 + 3x + 2) / (x2 + 5x + 6))
factor and combine numerators and denominators
simplify:
what values can x not equal?
are they holes or asymptotes?
factor and combine: ((x + 3)(x + 1)(x + 2)) / ((5(x + 4)(x + 3)(x + 2))
simplify: (x + 1) / 5(x + 4)
x can't equal -4, -3, -2
-4 asymptote, -3, -2 holes
y = 5 / x
Is this direct or inverse relationship?
What is the value of k?
inverse
5
The distance needed to stop a car varies directly as the square of its speed. It requires .12 km to stop a car at 70 km/h.
Write the equation for distance
What is the value of k?
What distance is required to stop a car at 80 km/h?
distance = k * speed2
k = .00002
distance = .157 km
((x + 3) / (5x + 20)) * ((x2 + 3x + 2) / (x2 + 5x + 6))
factor and combine numerators and denominators:
simplify:
x-restrictions:
factor and combine: ((x + 3)(x + 2)(x + 1)) / ((5(x + 4)(x + 3)(x + 2))
simplify: (x + 1) / 5(x + 4)
x can't equal -4, -3, -2
((x - 5) / 6) / ((2x - 10) / 12)
flip:
factor and combine numerators and denominators
simplify:
x-restrictions
flip: ((x - 5) / 6) * ((12) / (2x - 10))
factor and combine: (12(x - 5)) / (12(x - 5))
simplify: 1
x can't equal 5
(15x2 / 45x3) / (5x6 / 9x4)
flip:
simplify:
what values can x not equal?
are those values holes or asymptotes
flip: (15x2 / 45x3) * (9x4 / 5x6)
simplify: 3 / 5x3
x can't equal 0
0 is an asymptote
y = 25x
Is this direct or inverse relationship?
What is the value of k?
direct
25
The number of hours required to do a job varies inversely as the number of people working. It takes 8 hours for 4 people to paint the inside of a house.
Write the equation for time
What is the value of k?
How long would it take 5 people to do the job?
time = k / people
k = 32
time = 6.4 hours
((x + 3) / (x2 - 4x + 4)) * ((2x2 - 2x - 4) / (x2 + 4x + 3))
factor and combine numerators and denominators
simplify:
x-restrictions:
factor and combine: (2(x + 3)(x - 2)(x + 1)) / ((x - 2)(x - 2)(x + 3)(x + 1))
simplify: 2 / (x - 2)
x can't equal 2, -3, -1
((x2 - 5x - 6) / (5x + 15)) / ((x2 - 3x - 4) / (7x + 21))
flip:
factor and combine numerators and denominators
simplify:
x-restrictions:
flip: ((x2 - 5x - 6) / (5x + 15)) * ((7x + 21) / (x2 - 3x - 4))
factor and combine: (7(x - 6)(x + 1)(x + 3)) / ((5)(x + 3)(x - 4)(x + 1))
simplify: (7(x-6)) / (5(x - 4))
x can't equal -3, 4, -1
((6x + 24) / (5x - 35)) * ((9x - 63) / (7x + 28))
factor and combine numerators and denominators
simplify:
what values can x not equal?
are those values holes or asymptotes?
factor and combine: (6(x + 4) * 9 (x - 7)) / (5 (x - 7) * 7 (x + 4))
simplify: 54 / 35
x can't equal -4 or 7
-4 and 7 are both holes