Direct, Inverse, & Joint Variation
We say…
→ “y varies directly as x”
→ “y is directly proportional to x”
→ “y is proportional to x”
Ex:
Ex:
y = 1.9x
y = −4x
2
y =
3
4
NOTE: For a Direct Variation k = _______
We say…
→ “y varies inversely as x”
→ “y is inversely proportional to x”
Ex:
Ex:
y =
13
x
π
y =
x
2
NOTE:
NOTE For an Inverse Variation k = _______
y =
20
3x
x
5
We say…
→ “z varies jointly as y and x”
→ “z is jointly proportional to x and y”
Ex:
2
3
z = xy , z = 4xy , z =
xy
7
Equations of some variation functions
y varies directly with x : _____________
y varies directly with the square of x : _____________
y varies inversely with x : _____________
y varies inversely with the cube of x : _____________
z varies jointly as a and b : _____________
z varies directly with the square of x and inversely with y : _____________
Ex:
Ex If y varies directly as x 2, and y = 8 when x = 2.
Equation:
Find y when x = 1
Ex:
Ex y is inversely proportional to x, and y = 0.4 when x = 2.5.
Equation:
Find y when x = 4
Ex:
Ex Tell whether the data shows a direct or indirect variation. If so, write an
equation relating x and y.
a)
b)
Ex: U varies jointly as V and the square of W. If V = 4 and W = 3, then U = 18.
Ex:
Ex According to Hooke’s Law, the force needed to stretch a spring is proportional
to the amount the spring is stretched. If fifty pounds stretches a spring five
inches, how much will the spring be stretched by a force of 120 pounds.
Equation:
A force of 120 pounds
Ex:
Ex The weight of a body varies inversely as the square of its distance from the
center of the earth. If the radius of the earth is 4000 miles, how much would
a 200 pound man weight 1000 miles above the surface of the earth?
Equation:
1000 miles above the earth’s surface
Ex: The frequency of a vibrating string varies inversely as its length. That is, a
longer string vibrates fewer times in a second than a shorter string. Suppose a
piano string 2 feet long vibrates 250 cycles per second. What frequency
would a string 5 feet long have?
Equation:
Frequency of a 5 foot long string: