# 3.4: Polynomials and Graphs

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

**1. Left and Right Behavior**

We will investigate the outer shape of several polynomials and explore the following rules:

Even | Odd | |

Pos | UU | DU |

Neg | DD | UD |

Where Even and Odd refers to the degree of the polynomial, Pos and Neg refers to the leading coefficient, And a U or a D refers to the left and right behavior of the curve.

Example 1

\[-3x^7 + 4x^4 - 1\]

has degree 7 which is odd and has leading coefficient -3 which is negative. Hence the left and right behavior is UD, i.e. the curve goes up on the left and down on the right.

**2. Max and Min**

Theorem

If \(f(x)\) is a polynomial of degree \(n\) then \(f(x)\) has at most \(n - 1\) relative extrema.

Where relative extrema are "lumps" of the graph.

Example 2

\[4x^5 + 2x^3 - x^2 + 7x + 12\]

has at most 4 relative extrema.

**3. Three Step Procedure For Graphing Polynomials**

**Step 1: **Factor the polynomial into linear factors of the form

\[ax + b\]

**Step 2 : **Determine the left and right behavior of the graph and the shape of the graph near each \(x\) intercept.

**Step 3: ** Connect the dots.

Example

Graph

\[y = x^4 - 10x^2 + 9.\]

1. We have

\[y = (x^2 -9)(x^2 - 1) = (x - 3)(x + 3)(x - 1)(x + 1).\]

2. The left behavior is up and the right behavior is up.

Near \(x = -3\) the graph is positive on the left and negative on the right.

Near \(x = -1\) the graph is negative on the left and positive on the right.

Near \(x = 1\) the graph is positive on the left and negative on the right.

Near \(x = 3\) the graph is negative on the left and positive on the right.

3.

Larry Green (Lake Tahoe Community College)

Integrated by Justin Marshall.