Equations and Graphs of Polynomial Functions

 

The equation of a polynomial function in the factored form is as follows :

 

F(x) = (x-a)(x-b)(x-c)(x-d) … etc

 

a, b, c, and d are the zeros/x-intercepts.

 

There are 3 cases for the x-intercept:

 

  1. i.e. (x-3)^1 or simply (x-3): in this case, the line will just go through the x-axis at this point without any change.
  1. i.e. (x-3)^2 : in this case, the line will touch the x-axis then bounce back (turning point). The x-axis would be tangent to the curve in this situation
  1. i.e. (x-3)^3 : in this case, the line will have a cubic shape when it goes through this point

 

Example:

 

In the graph below, at x=3 the line goes through because it’s linear, at x=5 the line bounces back because it is squared, and at x=9 the line takes a cubic form. Since the highest degree is 6 (even), this function extends from quadrant 2 to quadrant 1. The reason the degree is 6, is because when you are multiplying terms, the exponents are added. In this case the exponents are 1, 2, and 3.

 

Equation: f(x) = (x-3)(x-5)(x-9)

 

 

 

 

If you have any questions, leave them in the comments below.