A Reform Approach to College Algebra
Marcel B. Finan
Arkansas Tech University
May 14, 2004
This supplement consists of my lectures of a freshmen-level College Algebra
class offered at Arkansas Tech University. The lectures are designed to
5 Geometric Properties of Linear Functions
In this section we discuss four geometric related questions of linear functions.
The first question considers the significance of the parameters m and b in the
equation f(x) = mx + b.
We have seen that the graph
Finding Input/Output of a Function
In this section we discuss ways for finding the input or the output of a function
defined by a formula, table, or a graph.
Finding the Input and the Output Values from a Formula
By evaluating a function, we mean figuring
Exponential Growth and Decay
Exponential functions appear in many applications such as population growth,
radioactive decay, and interest on bank loans.
Recall that linear functions are functions that change at a constant rate. For
example, if f(x) = mx +
Average Rate of Change and Increasing/Decreasing Functions
Now, we would like to use the concept of the average rate of change to test
whether a function is increasing or decreasing on a specific interval. First,
we introduce the following definition: We
Exponential Functions Versus Linear Functions
The first question in this section is the question of recognizing whether a
function given by a table of values is exponential or linear. We know that for
a linear function, equal increments in x correspond to
3 Linear Functions
In the previous section we introduced the average rate of change of a function.
In general, the average rate of change of a function is different on different
intervals. For example, consider the function f(x) = x2. The average rate of
In general, data obtained from real life events, do not match perfectly simple
functions. Very often, scientists, engineers, mathematicians and business
experts can model the data obtained from their studies, with simple linear
The Effect of the Parameters a and b
Recall that an exponential function with base a and initial value b is a function
of the form f(x) = b ax. In this section, we assume that b > 0. Since
b = f(0) then (0, b) is the vertical intercept of f(x). In this se
The value of a new computer equipment is $20,000 and the value drops at a
constant rate so that it is worth $ 0 after five years. Let V (t) be the value
of the computer equipment t years after the equipment is purchased.
(a) Find the slope m a
Rate of Change and Concavity
We have seen that when the rate of change of a function is constant then its
graph is a straight line. However, not all graphs are straight lines; they may
bend up or down as shown in the following two examples.
Piecewise Defined Functions
Piecewise-defined functions are functions defined by different formulas
for different intervals of the independent variable.
Example 9.1 (The Absolute Value Function)
(a) Show that the function f(x) = |x| is a piecewise defined
Let x and y be two quantities related by the equation
x2 + y2 = 4.
(a) Is x a function of y? Explain.
(b) Is y a function of x? Explain.
(a) For y = 0 we have two values of x, namely, x = 2 and x = 2. So x is
not a function of y.