Date:
Section 2.6
Digital Notes
College Algebra
Constructing Functions with Variation
Direct Variation (y  kx)
Example - Suppose you are driving at a constant 60mph down the highway.
t
d
Definition - y varies directly as x if changing x by a certain factor causes y to change
by the same factor.
If y varies directly as x, then y  kx, where k is the constant of proportionality.
Note - The phrase ’varies directly as’ has the same meaning as ’directly proportional
to.’
Recall - Even though they are both represented by letters, there is a significant
difference between a variable and a constant. A variable is an unknown number that
is subject to change (or vary). A constant is an unknown number that is fixed and will
not change within a given problem. The slope, m, is an example of a constant.
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Inverse Variation (y 
k
x
)
Example - Suppose you must drive 300 miles for a meeting.
r
t
Definition - y varies inversely as x if changing x by a certain factor causes y to change
by the reciprocal of that factor.
If y varies inversely as x, then y 
k
x
, where k is the constant of proportionality.
Note - The phrase ’varies inversely as’ has the same meaning as ’inversely
proportional to.’
Example - 1. Your grade on the next test, G, varies directly with the number of hours,
n, that you study for it.
Example - 3. The volume of a gas in a cylinder, V, is inversely proportional to the
pressure on the gas, P.
Now try 2, 4.
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Example - 23. T is inversely proportional to y, and T  −30 when y  5.
Example - 25. m varies directly as the square of t, and m  54 when t  3 2 .
Now try 22, 24, 26. (More practice? 21)
Example - 31. If P is inversely proportional to w, and P 
when w  16 ?
2
3
when w 
1
4
, what is P
Now try 30, 32. (More practice? 29)
Varies Jointly (y  kxz)
Definition - y varies jointly as x and z means that y varies directly with both x and z at
the same time.
If y varies jointly as x and z, then y  kxz, where k is the constant of proportionality.
Note - The phrase ’varies jointly as’ has the same meaning as ’jointly proportional to.’
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Example - 33. If A varies jointly as L and W, and A  30 when L  3 and W  5 2 ,
what is A when L  2 3 and W  12 ?
Now try 6, 20, 28, 34. (More practice? 5, 33).
Combined Variation
Rule - When y varies with multiple variables, remember the following:
1. There is only ever one constant k and it is always in the numerator.
2. If y varies directly with a variable, that variable goes in the numerator (multiplied to
whatever is already there).
3. If y varies inversely with a variable, that variable goes in the denominator
(multiplied to whatever is already there).
Example - 27. y varies directly as x and inversely as the square root of z, and y  2. 192
when x  2. 4 and z  2. 25.
Now try 8, 36. (More practice? 7, 19, 35)
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Word Problems
Example - 63. Calvin believes that his grade on a college algebra test varies directly
with the number of hours spent studying during the week prior to the test and inversely
with the number of hours spent at the Beach Club playing volleyball during the week
prior to the test. If he scored 76 on a test when he studied 12 hours and played 10
hours during the week prior to the test, then what score should he expect if he studies
9 hours and plays 15 hours?
Now try 54, 56, 58. (More practice? 53, 55, 57).
Don’t forget to do 38, 40, 42.
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