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# 3.4: Polynomials and Graphs

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### 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.