1. Use the following equation to perform the operations below:

$y = (frac{x + 5}{x + 1} div frac{(x + 3)(x - 2)}{(x - 4)(x + 1)}) - frac{1}{x - 2}$

a. Graph the original (not a reduced version) equation using a graphing calculator or an on-line graphing tool like the one at Where are the vertical asymptotes of the graph located? Explain how you can use the equation to verify these vertical asymptotes.
b. What graphical feature occurs at $footnotesize{x} = -1$? Use the “trace” function, zoom in on the graph, or look at a table of values to verify what is happening at $footnotesize{x} = -1$. Why isn’t there a vertical asymptote at $footnotesize{x} = -1$?
c. Solve the equation for $footnotesize{x}$ when $footnotesize{y = 0}$ (find the roots of the equation). Show all of your work and explain the strategies you use in each step.

2. The reflector of a satellite dish is in the shape of a parabola with a diameter of 4 feet and a depth of 2 feet. To get the maximum reception we need to place the antenna at the focus.

a. Write the equation of the parabola of the cross section of the dish, placing the vertex of the parabola at the origin. Convert the equation into standard form, if necessary. What is the defining feature of the equation that tells us it is a parabola?
b. Describe the graph of the parabola. Find the vertex, directrix, and focus.
c. Use the endpoints of the latus rectum to find the focal width.
d. How far above the vertex should the receiving antenna be placed?

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Originally posted 2022-07-12 15:40:01.