Polynomials and Graphs

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Polynomials and Graphs

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 $Graph through the origin that goes down steeply on the left and up steeply on the right.$

-3x⁷ + 4x⁴ - 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.

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:

4x⁵ + 2x³ - x² + 7x + 12

Has at most 4 relative extrema.

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 = x⁴ - 10x² + 9
1. We have
  
  y = (x² -9)(x² - 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. $Graph that goes down then up then down then back up.$

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