Unit 8
MHF 4U1
Lesson 3– Rates of Change in Polynomial and Rational Functions
The methods used to calculate the average rate of change and the instantaneous rate of
change can be used for both polynomial and rational functions.
Recall the following:
Average Rate of Change =
f ( x2 )  f ( x1 )
which is the slope of the secant line that
x2  x1
passes through the graph of a function.
f ( a  h)  f ( a )
at the point x = a and where h is a
h
very small value (  0.1 ,  0.01 ,  0.001 , etc.). This is the slope of the tangent line to the
Instantaneous Rate of Change =
graph of the function.
Polynomial Functions
The instantaneous rate of change of a polynomial function at any of its turning points is
0.
Example 1: For the function f ( x)  ( x  3) 3  1 determine the following:
a) The average rate of change on the interval 2  x  5 .
b) The instantaneous rate of change at x  3 .
Unit 8
MHF 4U1
Rational Functions
You cannot determine the average and instantaneous rates of change at a point where
the graph has a hole or vertical asymptote.
The instantaneous rate of change at a vertical asymptote is undefined.
The instantaneous rate of change at points that are approaching a vertical asymptote
become very large positive or very large negative values.
The instantaneous rate of change near a horizontal asymptote approaches zero.
Example 2: For the function
x2
determine the following:
x 1
a) The average rate of change over the interval 2  x  7 .
b) The instantaneous rate of change at x = 2.