Download Direct and Inverse Variation Objectives

Objectives
Direct and Inverse Variation
Direct Variation
– Solve direct variation problems.
– Solve inverse variation problems.
– Solve combined variation problems.
– Solve problems involving joint variation.
Direct variation COULD involve an
nth power of x (no longer linear)
• y varies directly as x (y is directly
proportional to x) if y = kx.
• k is the constant of variation (constant of
proportionality)
• Still involves a constant
of proportionality
n
• This is the graph of a linear function with
slope = m, crossing through the origin.
• y is directly proportional to the nth power of x.
y = kx
• Example: You are paid $8/hr. Thus, pay is
directly related to hours worked:
pay=8(hours worked)
Solving Variation Problems
1. Write an equation that describes the
given English statement.
2. Substitute the given pair of values into
the equation in step 1 and solve for k, the
constant of variation.
3. Substitute the value of k into the
equation in step 1.
4. Use the equation in step 3 to answer the
problem’s question.
Example
• An object’s weight on the moon, M, varies directly
as its weight on Earth, E. Neil Armstrong, the first
person to step on the moon on July 20, 1969,
weighed 360 pounds on Earth (with all his
equipment on) and 60 pounds on the moon. What
is the moon weight of a person who weighs 186
pounds on Earth?
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Inverse Variation
Example
• The water temperature of the Pacific Ocean
varies inversely as the water’s depth. At a depth
of 1000 meters, the water temperature is 4.4°
Celsius. What is the water temperature at a
depth of 5000 meters?
k
y=
x
• y varies inversely as x or y is inversely
proportional to x.
• k is the constant of variation.
• As x gets bigger, y gets smaller
• As x gets smaller, y gets bigger
Combined Variation Problem
• y is impacted by TWO variables in TWO different
ways. One variable causes y to get bigger, while
the other variable causes it to become smaller.
y=
k⋅x
z
Example
• y varies jointly as m and the square of n and
inversely as p.
y = 15 when m = 2, n = 1, and p = 6.
Find y when m = 3, n = 4, and p = 10.
y varies directly as x
and inversely as z
• As x gets bigger, y gets bigger, but as z gets
bigger, y gets smaller.
• k must take into account both influences
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