Graphing Polynomial Functions

 Hi, my name is Anton, hopefully Mr P had a fantastic birthday weekend :)

Today I will be showing you guys how to graph polynomial functions. There are a few characteristics for graphing polynomial functions. The type of function and the degree (even or odd), the end behavior, number of x and y intercepts, the maximum or minimum value.

To sketch the graph of  polynomial function, use the x and y intercepts, the degree of the function, and the sign of the leading coefficient.

The x intercepts are the roots of the functions.

Determine the zeros (x intercepts) of a polynomial function from the factors.

Use the factor theorem to express a polynomial function in factored form.

For Graphing Odd Degree Polynomials
Factor the equation and look for the x intercept

If the leading coefficient is positive, the left arm will go down and the right arm will go up.

If the leading coefficient is negative, the left arm will go up and the right arm will go down.

Plot the zeros and the y intercept on the graph and draw the line the connects the end behavior, zeros, and y intercept.

To look for the y intercept, simply multiply the numbers in the bracket.

In this case (x+2)(x-1)(x-3) ..... (2)(-1)(-3)=6

So the y intercept is 6

For Graphing Even Degree Polynomials
Factor the equation
In this case its already factored out.

y=(x+1)(x-4)(x+5)(x-2)

If the leading coefficient is positive, both arms will go up.

If the leading coefficient is negative, both arms will go down. 

Plot the zeroes and the y intercept on the graph and draw the line that connects the end behavior, zeros and y intercept.

 (1)(-4)(5)(-2)=40

 So the y intercept is 40

Important graphing rules

1. Each root of the equation is unique (i.e. There are no two or more of the same root) meaning there's only one root. The curve crosses at the x-axis at this point

2. If you have two or more of the same root:
 a) An EVEN number of the same root, the curve is tangent meaning it will bounce off at the x-axis.
      For example: (x+1)², (x-3)⁶, (x-4)⁸

 b) An ODD number of the same root, the curve will cross at the x-axis
      For example: (x-3)³, (x+2)⁷, (x+5)⁹

3. The higher the degree of multiple roots, the more the graph flattens out at the point

Here are some examples:

Hopefully everyone had a good long weekend and once again happy belated to Mr P (: