College Algebra
Answer/Discussion to Practice Problems
Tutorial 35: Graphs of Polynomial Functions
Answer/Discussion
to 1a

Do you think that the graph rises or falls to the left and to the
right?
Since the degree of the polynomial, 4, is even and the leading
coefficient, 3, is negative, then the graph of the given polynomial falls
to the left and falls to the right. 

*Setting the 1st factor = 0
*Solve for x
*x = 0 is a zero 
Since the exponent on this factor is 2, then the multiplicity for
the zero x = 0 is 2.
Since the multiplicity is 2, which is even, then the graph touches
the xaxis and turns around at the zero x = 0. 

*Setting the 2nd factor = 0
*Solve for x
*x = 3 is a zero 
Since the exponent on this factor is 2, then the multiplicity for
the zero x = 3 is 2.
Since the multiplicity is 2, which is even, then the graph touches
the xaxis and turns around at the zero x = 3. 
The yintercept is (0, 0). 
Step 4: Determine if
there is any symmetry. 
It is not symmetric about the yaxis. 

*Plug in x for x
*Take the opposite of f(x)

It is not symmetric about the origin. 
Since the degree of the function is 4, then there is at most 4 
1 = 3 turning points. 
To get a more accurate curve, lets find some points that are in between
the points we found in steps 2 and 3: 
x


(x, y)

2


(2, 12)

1


(1, 12)

Answer/Discussion
to 1b

Do you think that the graph rises or falls to the left and to the
right?
Since the degree of the polynomial, 3, is odd and the leading coefficient,
1, is positive, the graph of the given polynomial falls to the left
and rises to the right. 

*Setting the 1st factor = 0
*Solve for x
*x = 0 is a zero 
Since the exponent on this factor is 2, then the multiplicity for
the zero x = 0 is 2.
Since the multiplicity is 2, which is even, then the graph touches
the xaxis and turns around at the zero x = 0. 

*Setting the 2nd factor = 0
*Solve for x
*x = 3 is a zero 
Since the exponent on this factor is 1, then the multiplicity for
the zero x = 3 is 1.
Since the multiplicity is 1, which is odd, then the graph crosses
the xaxis at the zero x = 3. 
The yintercept is (0, 0). 
Step 4: Determine if
there is any symmetry. 
It is not symmetric about the yaxis. 

*Plug in x for x
*Take the opposite of f(x)

It is not symmetric about the origin. 
Since the degree of the function is 3, then there is at most 3 
1 = 2 turning points. 
To get a more accurate curve, lets find some points that are in between
the points we found in steps 2 and 3: 
x


(x, y)

1


(1, 2)

2


(2, 4)

Last revised on March 14, 2012 by Kim Seward.
All contents copyright (C) 2002  2012, WTAMU and Kim Seward. All rights reserved.

