9.1 Direct, Inverse, & Joint Variation
Direct Variation
Two variables x and y show direct variation provided y = kx and k ≠ 0. The
nonzero constant k is called the constant of variation and y is said to “vary
directly with x.” The graph of y = kx is a line through the origin.
Ex: The variables x and y vary directly and y = 12 when x = 4 .
a.) Write an equation that relates x and y.
b.) Find the value of y when x = 10.
Ex: Tell whether the data show direct variation. If so, write an equation relating
x and y.
Direct variation? _________________
If yes, what’s the equation? _________________
If yes, find the price for a 28 inch gold chain. ___________
Direct variation? _________________
If yes, what’s the equation? _________________
If yes, find the price for .8 carats of loose diamonds._________
Inverse Variation
Two variables x and y show inverse variation if they are related as follows:
y=
k
,k ! 0
x
As above, the nonzero constant k is called the constant of variation and y is
said to “vary inversely with x.”
Ex: The variables x and y vary inversely, and y = 8 when x = 3 .
a.) Write an equation that relates x and y.
b.) Find the value of y when x = - 4.
Ex: Tell whether x and y show direct variation, inverse variation, or neither.
Given Equation
a)
Rewritten Equation
Type of Variation
y
=x
5
b) y = x + 2
c) xy = 4
Word Problems
The speed of the current in a whirlpool varies inversely with the distance from the
whirlpool’s center. The Lofoten Maelstrom is a whirlpool located off the coast of Norway.
At a distance of 3 kilometers (3000 meters) from the center, the speed of the current is
about 0.1 meter per second. Describe the change in the speed of the current as you
move closer to the whirlpool’s center.
The table below compares the wing flapping rate r (in beats per second) to the wing
length l (in centimeters) for several birds. Do these data show inverse variation? If so,
find a model for the relationship between r and l.
Inverse variation?
______________
If yes, what’s the equation?
______________
Joint Variation
Joint variation occurs when a quantity varies directly as the product of two or
more other quantities. For instance, if z = kxy where k ≠ 0, then z varies
ky
jointly with x and y. Additionally, if z =
, z varies directly with y and
x
inversely with x.
Ex: z varies jointly with x and y. Given that x = 3, y = 8 and z = 6.
a.) Write an equation relating x, y, and z.
b.) Then find z when x = -2 and y = 4.
Ex: Write an equation for the given relationship.
a)
x varies inversely with y and directly with z.
__________________
b)
y varies jointly with z and the square root of x.
__________________
c)
w varies inversely with x and jointly with y and z.
__________________
d) y varies inversely with the square of x.
__________________
e) z varies directly with y and inversely with x.
__________________
Homework: p. 537 – 538 # 21 - 43 odd, 54