L
inear
F
unctions
In single-variable calculus, we studied linear equations because they allowed us to
approximate complicated curves with a fairly simple linear function. Our goal will be to
use linear functions in two dimensions to approximate complicated surfaces.
A linear function in one variable could be written in point-slope form. That is, we said
that
y
=
y
0
+
m
(
x
–
x
0
). Once we knew a point on the linear and the slope, we had all the
information necessary to specify the equation.
When dealing with two variables, we have a similar point-slope form. This time,
however, we have two slopes: one in the
x
-direction and another in the
y
-direction.
Equation of a Linear Function
If a linear function (a plane) passes through the point (
x
0
,
y
0
,
z
0
) and has slope
m
in the
x
-direction and slope
n
in the
y
-direction, the equation of the linear
function (plane) is given by:
z
=
f
(
x
,
y
) =
z
0
+
m
(
x
–
x
0
) +
n
(
y
–
y
0
)
Letting
c
=
z
0
–
mx
0
–
ny
0
, we can also write
f
(
x
,
y
) in the form
z
=
f
(
x
,
y
) =
c
+
mx
+
ny
Just as two points uniquely determine the equation of a line in 2-space, three points
uniquely determine the equation of a plane in 3-space.
Example 1:
Find the equation of the plane passing through the points (1, 2, 3), (1, 4, 1), (2, 2, 5).
Solution:
Notice that the first two points have the same
x
-value. So, we can use them to find the
slope of the plane in the
y
-direction. As
y
changes from 2 to 4, the
z
-value changes from 3
to 1. Thus, the slope in the
y
-direction is given by
n
=
D
z
/
D
y
= (1 – 3)/(4 – 2) = -1.