SP#2: Unit E Concept 7 - Graphing a polynomial and identifying all key parts

Step by step: 1) Factor out the x.
2) Factor the rest of the equation.
3) List the end behavior (learned in concept 4)
4) List the x-intercepts (the zeroes) with their multiplicities (concept 6).
5) Find the y-intercept.
6) If possible find the extremas (the min and max) and then list in intervals of increase and decrease.
7) Plot all available points onto the graph, indicate your end behavior and then draw the graph following all graphing rules.

 

 

       This is an example of a factorable polynomial being graphed with all the appropriate parts. There are five steps that will always be there. Those are factoring the equation, end behavior, x-intercepts, y-intercept, and plotting/graphing them all. The parts that will sometimes be there are those of the extremas and the intervals of increase and decrease. This problem is to show you what these kind of problems look like and also what they consist of. They are multistep equations that involve your full attention. They are exciting problems that really review everything that we have been learning and introducing us to future concepts. They are great problems.

     This problem is not easy peasy though. You cannot breeze through these problems. You really have to pay attention to the end behaviors and the zeroes. The end behavior is based off of the highest degree and the leading coefficient in the standard equation. This one happens to be an even (the degree) positive (the leading coefficient). That means it will go up on both the left and right of the graph. Other equations will see different end behaviors though. Make sure your end behaviors are going in the correct direction and do not make up intercepts to keep in line with your end behaviors. The second is the zeroes. You have to make sure to put their multiplicities. That tells you how they will appear on the graph. If it is a single multiplicity then it will go straight through the x-axis, if it is a multiplicity of two then it will bounce on the x-axis, and if it is a multiplicity of three then it curves through the x-axis. These intercepts are the only times the graph will cross the x-axis. It will never go past a solid line with no intercept. Those are really the only tricky parts.