(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.
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- 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.
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