Unit 12.3 Variation
Direct Variation
A variable quantity y varies directly as (or is directly proportional to) a variable x. If there is a constant “k” such that y = kx. The constant “k” is often referred to as the “constant of variation”. With direct variation, x will always be in the numerator.
Direct Variation
y = k⋅x
Direct Variation
If y varies directly as x, and y = 6 when x = 2, find y is x = 6.
y = kx
6 = k ⋅ 2
Solve for k
k = 3
The value of k will stay the same no matter what x and y values you plug in.
y = 3x
Find y when x = 6
y = 3 ⋅ 6
y = 18
Inverse Variation
A variable quantity y varies inversely as (or is inversely proportional to) a variable x. If there is a constant “k” such that [latex]\ y=\frac{k}{x}[/latex]. The constant “k” is often referred to as the “constant of variation”. With inverse variation, x will always be in the denominator.
Inverse Variation
[latex]\ y=\frac{k}{x}[/latex]
Inverse Variation
If y varies inversely as the cube of x, and y = -1 when x = 3, find y if x = -3.
[latex]\ y = \frac{k}{x^3}[/latex]
[latex]\ -1 = \frac{k}{3^3}[/latex]
Solve for k
[latex]\ -1 = \frac{k}{27}[/latex]
k = -27
The value of k will stay the same no matter what x and y values you plug in.
[latex]\ y = \frac{-27}{x^3}[/latex]
Find y when x = -3
[latex]\ y = \frac{-27}{-3^3}[/latex]
[latex]\ y = \frac{-27}{-27}[/latex]
y = 1
Combined Variation
If a variable varies either directly or inversely with more than one other variable, the variation is said to be combined variation. If the combined variation is all direct variation (the variables are multiplied), then it is called joint variation.
Combined Variation
If z varies jointly as x2 and y, and z = 18 when x = 2 and y = 4, what is z when x = 4 and y = 3?
Write the full formula
z = k ⋅ x2 ⋅ y
18 = k ⋅ 22 ⋅ 4
Solve for k
18 = k ⋅ 16
k = [latex]\frac{9}{8}[/latex]
The value of k will stay the same no matter what x and y values you plug in.
z = [latex]\frac{9}{8}[/latex] ⋅ x2 ⋅ y
Solve for z when x = 4 and y = 3
z = [latex]\frac{9}{8}[/latex] ⋅ 42 ⋅ 3
z = 54