Name ____________________________________Date ______________Block _____
Adv. Math: Chapter 3
Lesson 8: Direct, Inverse, and
Joint Variation
Objectives:
 Solve problems involving direct, inverse, and joint variation
Direct Variation:
y varies directly as 𝒙𝒏
if there is some nonzero constant k such that
𝒚 = 𝒌𝒙𝒏 , 𝒏 > 𝟎, 𝒌 is called the constant of variation
**As one goes up, the other goes up: speed and braking distance
Ex. 1a: Suppose y varies directly as x and y = 27 when x = 6.
a. find the constant of variation and write the equation of the form 𝒚 = 𝒌𝒙𝒏
b. use the equation to find the value of y when x = 10
Ex. 1b. Suppose y varies directly as x and y = 45 when x = 2.5
c. find the constant of variation and write the equation of the form 𝒚 = 𝒌𝒙𝒏
d. use the equation to find the value of y when x = 4
Using proportions . . .
𝒚𝟏 = 𝒌𝒙𝒏𝟏
𝒂𝒏𝒅 𝒚𝟐 = 𝒌𝒙𝒏𝟐
→
Ex. 2a: If y varies directly as the cube of x and y = -67.5 when x = 3, find x
when y = -540.
Ex. 2b: If y varies directly as the square of x and y = 30 when x = 4, find x
when y = 270.
Inversely Proportional (Inverse Variation):
y varies inversely as xn
if there is some nonzero constant k such
that 𝒙𝒏 𝒚 = 𝒌 𝒐𝒓 𝒚 =
𝒌
𝒙𝒏
, 𝒏>𝟎
**As one goes up, the other goes down: elevation and air temperature
Ex. 3a: If y varies inversely as x and y = 21 when x = 15, find x when y = 12.
Ex. 3b: If y varies inversely as x and y = 14 when x = 3, find x when y = 30.
Joint Variation:
y varies jointly as xn and zn if there is some nonzero constant k such
that 𝒚 = 𝒌𝒙𝒏 𝒛𝒏 , 𝐰𝐡𝐞𝐫𝐞 𝒙 ≠ 𝟎, 𝒛 ≠ 𝟎, 𝐚𝐧𝐝 𝒏 > 𝟎.
Ex. 4: Find the constant of variation for the following relation and use it to write an
equation for the statement. Then find the stated value.
If y varies jointly as x and the cube of z and y = 16 when x = 4 and z = 2, find
y when x = -8 and z = -3.