College Algebra Lecture Notes
Section 3.4
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Section 3.4: Graphing Polynomial Functions
Big Idea: Some general knowledge about polynomials enables you to sketch any polynomial
reasonably quickly and accurately.
Big Skill: You should be able to sketch a given polynomial with accurate end behavior and
intercepts.
A. Identifying the Graph of a Polynomial Function
Polynomial Graphs and Turning Points
 If P  x  is a polynomial function of degree n, then the graph of P has at most n – 1
turning points. A turning point is where the graph turns from increasing to decreasing;
i.e., an extreme value.
 If the graph of a function P has n – 1 turning points, then the degree of P  x  is at least n.
In addition:
1. A polynomial of degree n will have at most n zeros.
2. The graph of a polynomial function is smooth (no sharp corners) and continuous.
Practice:
1. Identify the number of turning points and zeros for the polynomial graphs below.
College Algebra Lecture Notes
Section 3.4
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B. The End Behavior of a Polynomial Graph
 The end behavior of a graph is what happens to the y values as the x values go toward
positive and negative infinity.
 For polynomials, the end behavior is that the y values will always tend toward either
positive or negative infinity.
 The leading term will always dominate as x goes to infinity, and so it determines the end
behavior.
The End Behavior of a Polynomial Graph
Given a polynomial P  x  with leading term ax n and n > 1:
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If n is even and a > 0, then the end behavior is that the y values increase without bound;
as x  , y   and as x  , y  
If n is even and a < 0, then the end behavior is that the y values decrease without bound;
as x  , y   and as x  , y  
If n is odd and a > 0, then the end behavior is that the y values are in opposite directions;
as x  , y   and as x  , y  
If n is odd and a < 0, then the end behavior is that the y values are in opposite directions;
as x  , y   and as x  , y  
Practice:
2. Verify these end behavior properties on the graphs in problem #1.
College Algebra Lecture Notes
Section 3.4
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C. Attributes of Polynomial Graphs with Zeroes of Multiplicity
Polynomial Graphs with Zeroes of Multiplicity
m
Given P  x  is a polynomial with factors of the form  x  c  with c a real number,
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If m is odd, the graph will cross through the x axis.
If m is even, the graph will bounce off the x axis (i.e., touch it at just one point).
In either case, the graph will be more compressed (flatter) near c for larger values of m
Practice:
3. State the zeros of the polynomial graph shown below and the minimum multiplicity of
each zero. Then state minimum possible degree of the polynomial and whether the
degree of the polynomial is even or odd.
4. State as much as you can about the graph of the polynomials
1
3
2
p  x    x  2  x  4  x  1 and q  x     x  2   x 2  x  1  x  4  .
2
College Algebra Lecture Notes
Section 3.4
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D. The Graph of a Polynomial Function
Guidelines for Graphing Polynomial Functions
P  x   an xn  an1 xn1   a1 x  a0
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Determine the end behavior of the graph.
Find the y-intercept (0, a0).
Find the zeroes using techniques from 3.3.
Use the y-intercept, end behavior, multiplicity of each zero, symmetry, and handcalculated midinterval points as needed to sketch a smooth, continuous curve
Practice:
5. Sketch the graph of the polynomial p  x   x3 13x  12 .
College Algebra Lecture Notes
Section 3.4
6. Sketch the graph of the polynomial q  x   x4  x3  20 x  16 .
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College Algebra Lecture Notes
Section 3.4
7. Sketch the graph of the polynomial p  x   x5  3x4  x3  3x2 .
E. Applications of Polynomials
Practice:
8. .
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