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Mathematics LibreTexts

3.4: Polynomials and Graphs

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.

Contributors

  • Integrated by Justin Marshall.