Download Section 5.1

MATH 1100
SECTION 5.1 Notes
Polynomial Function & Their Graphs – Text Pages 311-323
Polynomial of Degree n:
Px   a n x n  a n 1 x n 1  ...  a1 x  a0
where an  0 .
The numbers an , an1 ,    , a1 , a0 are called the coefficients of the
polynomial.
a0 is called the constant term
an is called the leading coefficient
a n x n is called the leading term
The graph of a polynomial is always smooth and continuous.
Guidelines For Graphing Polynomial Functions:
1) ZEROS:
Find all x-intercepts.
2) Find the y-intercept
3) Make a TABLE using the x-intercepts as the “important” (critical)
points  determine whether the graph is positive or negative in
these intervals.
4) Use the leading coefficient to determine the end behavior of P(x).
5) GRAPH:
Plot the intercepts and other points found in the table
 sketch a smooth curve through these points that exhibits the
proper end behavior.
Before using these guidelines, we must discuss End Behavior & Zeros:
End Behavior of the graph of a Polynomial Function:
The end behavior of a polynomial is a description of what
happens as x becomes large in the positive or negative direction.
Notation:
x   means “x becomes large in the positive direction”
x   means “x becomes large in the negative direction”
For any polynomial, the end behavior is determined by the leading term.
Zeros of Polynomials:
If P is a polynomial function and if c is a number such that
Pc  0 , then we say that c is a zero of P.
The following are equivalent ways of saying this same thing:
1.)
2.)
3.)
c is a zero of P.
x  c is a root of the equation Px   0 .
x  c is a factor of Px .
Example 1:
Sketch Px   x 4  3x 3  4 x 2
STEP 1:
Find all x-intercepts (zeros)
STEP 2:
Find the y-intercept
STEP 3:
Make a table (or T-Chart)
STEP 4:
Determine End Behavior
STEP 5:
Sketch the graph
Example 2:
Sketch Px   x 3  2 x 2  4 x  8
STEP 1:
Find all x-intercepts (zeros)
STEP 2:
Find the y-intercept
STEP 3:
Make a table
STEP 4:
Determine End Behavior
STEP 5:
Sketch the graph
Example 3:
Sketch Px  x  3x  23x  2
STEP 1:
Find all x-intercepts (zeros)
STEP 2:
Find the y-intercept
STEP 3:
Make a table
STEP 4:
Determine End Behavior
STEP 5:
Sketch the graph
Example 4:
Sketch Px   x 4  3x 2  4
STEP 1:
Find all x-intercepts (zeros)
STEP 2:
Find the y-intercept
STEP 3:
Make a table
STEP 4:
Determine End Behavior
STEP 5:
Sketch the graph