1. 1
PRECALCULUS I
Dr. Claude S. Moore
Danville Community College
Mathematical Modeling
•Direct, inverse, joint variations;
Least squares regression
2. 1. y varies directly as x.
2. y is directly proportional to x.
3. y = kx for some nonzero constant m.
NOTE: k is the constant of variation or the
constant of proportionality.
Example: If y = 3 when x = 2, find k.
y = kx yields 3 = m(2) or m = 1.5.
Thus, y = 1.5x.
Direct Variation Statements
3. 1. y varies directly as the nth power of x.
2. y is directly proportional to the nth
power of x.
3. y = kxn for some nonzero constant k.
NOTE: k is the constant of variation or
constant of proportionality.
Direct Variation as nth Power
4. 1. y varies inversely as x.
2. y is inversely proportional to x.
3. y = k / x for some nonzero constant k.
NOTE: k is the constant of variation or the
constant of proportionality.
Example: If y = 3 when x = 2, find k.
y = k / x yields 3 = k / 2 or k = 6.
Thus, y = 6 / x.
Inverse Variation Statements
5. 1. z varies jointly as x and y.
2. z is jointly proportional to x and y.
3. z = kxy for some nonzero constant k.
NOTE: k is the constant of variation.
Example: If z = 15 when x = 2 and y = 3,
find k.
z = kxy yields 15 = k(2)(3) or k = 15/6 = 2.5.
Thus, z = 2.5xy.
Joint Variation Statements
6. BICYCLING A bicyclist tips
the bicycle when making turn.
The angle B of the bicycle
from the vertical direction is
called the banking angle.
USING DIRECT AND INVERSE VARIATION IN REAL LIFE
Writing and Using a Model
banking angle, B
7. Writing and Using a Model
From the graph, you can see that
B = 32° when r = 3.5 feet.
B =
k
r
32 =
k
3.5
112 = k
SOLUTION
Turning Radius
Banking
angle
(degrees)
Write direct variation model.
Substitute 32 for B and 3.5 for r.
Solve for k.
Find an inverse variation model
that relates B and r.
The model is B = , where B is in degrees and r is in feet.
112
r
8. Writing and Using a Model
SOLUTION
Use the model to find the banking
angle for a turning radius of 5 feet.
Substitute 5 for r in the model you just found.
B =
112
5
= 22.4
Turning Radius
Banking
angle
(degrees
)
When the turning radius is 5 feet, the banking angle is about 22°.
