MTHSC 360 – Working with Polynomials
January 15, 2008
1 Three ways to compute a straight line.
In High School Algebra, we learned that the equation for a straight line was given by the
expression
y
=
mx
+
β
(1)
where
m
is the slope (the rise over the run) of the straight line and
β
is the
y
intercept. This
is called the
slopeintercept
form for the equation of a straight line. This is a particularly
convenient form if we know both the
y
intercept, (0
,y
0
), and the
x
intercept, (
x
0
,
0) because
β
=
y
0
and
m
=

β/x
0
.
Then in Calculus we began to make heavy use of the
pointslope
form for the equation of a
straight line.
y
=
m
(
x

a
) +
f
(
a
)
(2)
First, given two points (
a,f
(
a
)) and (
b,f
(
b
)),
m
is the slope of the secant line, (the straight
line through the two points).
m
=
f
(
b
)

f
(
a
)
b

a
Then in the limit as
b
got closer and closer to
a
, we got the equation for the tangent line.
y
=
f
0
(
a
)(
x

a
) +
f
(
a
)
where
m
=
f
0
(
a
). It’s important to note that if the function,
f
(
x
), is in fact a straight line,
then we have
m
=
f
0
(
a
) =
f
(
b
)

f
(
a
)
b

a
and
β
=
bf
(
a
)

af
(
b
)
b

a
However, the big advantage of the pointslope form is that it focuses our attention on the
neighborhood of the point (
a,f
(
a
)) and emphasizes how things change with
x
– especially
for
x
near
a
. In other words, when
x
is close to
a
, then
f
(
x
) is close to
f
(
a
) and the exact
value of
f
(
x
) is obtained by adding the small correction
m
(
x

a
).
Of course we know from geometry that two points determine a straight line, and this leads
to a third form of the equation of a straight line given two points, (
a,f
(
a
)) and (
b,f
(
b
)).
This form is called the
twopoint
form.
y
=
f
(
a
)
x

b
a

b
+
f
(
b
)
x

a
b

a
(3)
Although this may seem like a strange way to write an equation for a straight line, it is easy
to see that (
x

b
)
/
(
a

b
) is a polynomial of degree 1 which has the value 1 at
x
=
a
and