Honors Precalculus module 8 project Show all steps taken and work needed to determine the needed equation and to answer each question. For those that require a sketch of the graph, be sure to label the x and y axes with appropriate increments, and label points on the graph. 1. The equation yz 4y 4x2 + 13 = 0 will produce a hyperbola. How can we tell by simply observing the equation? In what directions do the branches of this hyperbola open? How do you know? Explain. Sketch a graph of this hyperbola, clearly indicating how you have determined the key characteristics (center, vertices, eccentricity, foci). Give the domain and range of this hyperbola. 2. Find the standard form of the equation of the ellipse satisfying the following conditions: endpoints of major axis (5, 4) and (3, 4); endpoints of minor axis (1, 1) and (1, 7). Graph this conic, marking the center and vertices. 3. Find the vertex, focus, and equation of the directrix for the conic given by (x + 1)2 = 20(y 3). Sketch a graph of this conic. True or False: Note: explaining your answer can either include a counterexample, a diagram, or a written explanation to support your answer. 4. True or False: The area of a circle with diameter d = 2r = 10 is greater than the area of an ellipse with major axis 2a = 10. Explain your answer. Note: the area of an ellipse is calculated with the formula Area = nab (with 'a' the length of the semi-major axis and 'h' the length of the semi- minor axis). (ac-8F 2 5. True or False: The equation + 0:51) = 1 is the equation of the ellipse with yertices (8, -6) and (8, 4), with a focus at (8, 2). Explain your