## Monday, September 16, 2013

### SP #2: Unite E Concept 7: Graphing polynomials, including: x -int, y-int, zeroes (with multiplicities), end behavior. All polynomials will be factorable.

(1) This problem will go over how to graph a polynomial and find the characteristics stated in the above title. In this particular problem, we will be using the quartic polynomial  x^4 - 4x^3 - 12x^2. In order to complete the problem, we will be going through a number of steps, including factoring the polynomial, finding end behavior, solving for the x and y intercepts, calculating extrema and intervals of increase/decrease, and of course graphing.

(2)The reader must pay attention to detail for a little mistake could ruin the entire graph. Also, one thing the reader must be attentive to is the set window when plugging the polynomial into a graphing calculator to graph. This is because the problem results in large numbers and as a result the graph does not fit into the standard zoom.
 First we factor out an x squared and factor the rest of the equation from there. The resulting factored equation is in the 2nd box of the picture above. The end behavior part is as it is because the highest degree is 4 which is even, and the leading coefficient is positive, so the graph is an even positive meaning both ends point up. To get the x- intercepts, you take each individual part with an 'x' from the factored equation and set them equal to zero. The zero of (0,0) is repeated because of the x squared so it has a multiplicity of 2. For the y- intercept you just plug in zero into x from the original equation. Maximums and minimums could be found using 2nd cal on your calculator and following the instructions. Next, find the intervals using the extrema. Plot the zeroes and extrema on a graph. Next, figure out how the graph goes through each zero. To do this, refer to the multiplicities. A multiplicity of 1 means the graph goes straight through the zero, whereas a multiplicity of 2 means the graph bounces off the zero. Since the end behavior is even positive, both ends of the graph point up. Now draw the graph and you will be finished.