Section 2.6
College Algebra
Mr. Faullin
Variation
Begin Class (5 minutes)
Attendance
Check in to class on your iPad.
Log Entry
Please update your time log.
Warm Up Problem
If you are only able to drive 60mph, then how long would it take you to
drive 300 miles?
Submit your answer in the ‘Short Answer’ box on the Student Response
webpage.
Break
Lecture
The Basics of Variation
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 variation constant.
Note: The phrase ‘varies directly as’ has the same meaning as ‘directly
proportional to.’
Example – Is k a variable?
What do you think? (1 minute)
TRUE.
k is a variable.
Answer:
FALSE.
k is not a variable.
Example – Suppose you are driving 60mph. Then the time you drive is
related to distance you drive in the following way:
t
1
2
3
4
6
d
60
120
180
240
360
The distance d varies directly with the time 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 variation constant.
Note: The phrase ‘varies inversely as’ has the same meaning as
‘inversely proportional to.’
Example – Suppose you must drive 300 miles for a meeting. Then your
speed is related to the time you drive in the following way:
r
10
20
30
40
60
t
30
15
10
7.5
The time t varies inversely with the speed r.
Example – Write the formula that describes this situation: Your grade
on the next test, G, varies directly with the number of hours, n, that you
study for it.
Example – Write the formula that describes this situation: The volume
of a gas in a cylinder, V, is inversely proportional to the pressure on the
gas, P.
What do you think? (1 minute)
A.
B.
k=
V
P
C.
D.
k=
Answer:
V = kP
P
V
V=
k
P
Example – T is inversely proportional to y, and
T = −30
Example – m varies directly as the square of t, and
Example – If P is inversely proportional to w, and
when
m = 54
P=
2
3
y = 5.
when
when
t=3 2.
w=
1
, find
4
the variation constant k.
What do you think? (3 minutes)
Type your answer in the ‘Short Answer’ box.
Answer:
What is P when
w=
1
?
6
Combined Variation
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
constant.
y = kxz , where
k is the variation
Note: The phrase ‘varies jointly as’ has the same meaning as ‘jointly
proportional to.’
Example – Write the formula that describes this situation: A varies
jointly as L and W.
What do you think? (2 minutes)
Type your answer in the ‘Short Answer’ box.
Answer:
If
A = 30
when
L=3
and W = 5
2 , what
is A when
L=2 3
and W = 1 ?
2
Combined Variation
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 – 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 . Find the formula and the constant
of variation.
Find the constant of variation k.
What do you think? (3 minutes)
Type your answer in the ‘Short Answer’ box.
Answer:
Application
Example – 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?
Break
What topic from this section did you understand least? (1 minute)
(Submit your selection using the ‘Multiple Choice’ box on the Student Response Webpage.)
A. Finding k and answering a question.
B. Direct vs Inverse Variation.
C. Combined Variation.
D. Word problems.
End