1. Arkansas Tech University
MATH 1113: College Algebra
Dr. Marcel B. Finan
4.1 Cubic and Quartic Polynomials
In addition to linear, exponential, logarithmic, and quadratic functions, many
other types of functions occur in mathematics and its applications. In this
section, we will study cubic functions and quartic functions.
Cubic Polynomials
A cubic polynomial is a polynomial of degree 3. That is, a function of the
form
f(x) = ax3
+ bx2
+ cx + d, a 6= 0.
The coefficient a is called the leading coefficient.
The graph of a cubic function has the following features:
(1) The graph is a continuous curve which means that it can be drawn without
picking up your pencil. There are no jumps or holes in the graph of a cubic
function.
(2) The graph is a smooth curve which means that there are no sharp turns
(like an absolute value) in the graph of the function.
(3) The y−intercept is the point (0, d).
(4) If a > 0 the right hand side of the graph will rise towards +∞ whereas
the left side will fall toward −∞. If a < 0 the right hand side of the graph
will fall towards −∞ whereas the left side will rise toward +∞.
A point on the graph where the curve changes from increasing to decreasing
or vice versa is called a turning point. A cubic function can have zero or two
turning points. A turning point where the graph is changing from increasing
to decreasing is called a maximum point. In the case, the graph is changing
from decreasing to increasing then the turning point is a minimum point.
If a maximum point is not the highest point on the graph then it is called a
local maximum. Otherwise, it is called an absolute maximum. Similarly,
if a minimum point is not the lowest point on the graph then it is called a
local minimum. Otherwise, it is called an absolute minimum.
Example 1
(a) Sketch the graph of the function
f(x) = −
1
3
x3
+
5
3
x2
− x − 3.
(b) Find the local maximum and local minimum if they exist.
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2. Solution.
The graph is given in Figure 4.1.1.
Figure 4.1.1
(b) Using the MAXIMUM and MINIMUM features of a graphing calculator
we find the local minimum at (1
3
, −256
81
) and a local maximum at (3, 0)
Quartic Polynomials A quartic polynomial is a polynomial of degree
4. That is, a function of the form
f(x) = ax4
+ bx3
+ cx2
+ dx + e, a 6= 0.
The coefficient a is called the leading coefficient.
The graph of a quartic function has the following features:
(1) The graph is a continuous curve which means that it can be drawn without
picking up your pencil. There are no jumps or holes in the graph of a cubic
function.
(2) The graph is a smooth curve which means that there are no sharp turns
(like an absolute value) in the graph of the function.
(3) The y−intercept is the point (0, e).
(4) If a > 0 then both the left hand side and the right hand side of the graph
will rise towards +∞. If a < 0 then both the left hand side and the right
hand side of the graph will fall towards −∞. (5) A quartic function can have
one or three turning points.
Example 2
Consider the quartic function given in Figure 4.1.2.
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3. Figure 4.1.2
State whether the leading coefficient is positive or negative.
Solution.
Since the graph is rising on both ends, the leading coefficient is positive
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