7.2 Reducing Rational Functions

630 Chapter 7 Rational Functions
The Secant Line
Consider the graph of the function f that we’ve drawn in Figure 3. Note that we’ve
chosen two points on the graph of f, namely (a, f(a)) and (x, f(x)), and we’ve drawn
a line L through them that mathematicians call the “secant line.”
y
(a, f(a))
(x, f(x))
L
f
x
a
x
Figure 3. The secant line passes through
(a, f(a)) and (x, f(x)).
The slope of the secant line L is found by dividing the change in y by the change in x.
Slope = ∆y f(x) − f(a)
=
∆x x − a
This slope provides the average rate of change of the variable y with respect to
the variable x. Students in calculus use this “average rate of change” to develop the
notion of “instantaneous rate of change.” However, we’ll leave that task for the calculus
students and concentrate on the challenge of simplifying the expression equation (23)
for the average rate of change.
◮ Example 24. Given the function f(x) = x 2 , simplify the expression for the
average rate of change, namely
f(x) − f(a)
.
x − a
First, note that f(x) = x 2 and f(a) = a 2 , so we can write
f(x) − f(a)
x − a
= x2 − a 2
x − a .
We can now use the difference of two squares pattern to factor the numerator and
cancel common factors.
x 2 − a 2
x − a
=
(x + a)(x − a)
x − a
= x + a
(23)
Version: Fall 2007