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Transcript
9.3 Ellipses 1. Describe the equation for an ellipse. How does it differ from the equation for a circle? 2. In the general equation for an ellipse (p.470), what does rx stand for? What does ry stand for? What do h and k stand for? Example 1. Following the example in the text, sketch the graph of the following ellipse 16x 2 + 96x + 4 y 2 −16y + 96 = 0 3. Write the geometric definition of an ellipse. 4. What are the foci? 5. Define focal radius. How are rx and ry related to the focal radius? Example 2. For the ellipse you sketched in example 1 find the foci and the focal radius. 6. Recall the general equation for a quadratic relation from 9.1. What would the coefficients look like if the relation was an ellipse? 9.4 Hyperbolas 1. How does the equation for a hyperbola differ from the equation for an ellipse? 2. Explain why the graph of a hyperbola has two branches. 3. How can you tell from the equation in which direction the hyperbola opens? 4. How do you determine the diagonal asymptotes of a hyperbola? 5. What is the x radius (rx) of a hyperbola? What is the y radius (ry)? 6. What is the transverse axis? What is the conjugate axis? 7. What are the equations of the asymptotes? 8. What are conjugate hyperbolas? 9. Write the geometric definition of a hyperbola. Example 1. Sketch the graph of −16x 2 + 64 x + 9y 2 − 54 y −127 = 0 Example 2: Find and plot the foci of the hyperbola in Example 1. 9.5 Parabolas 1. Review the general forms of equations for parabolas. What does standard form look like? What does factored form look like? What does vertex form look like? 2. What is the formula for the x coordinate of the vertex? How do you find the y coordinate of the vertex? Example (review) For the following parabola find the vertex, y intercept, symmetric point, x intercepts, and draw the graph. y = 3x 2 + 4 x − 4 4. Consider the general equation for a quadratic relation. How does the equation for a parabola differ from the equation for a hyperbola and the equation for an ellipse? 5. How does a parabola with a y2 term differ from a parabola with an x2 term? Example. Graph the following parabola. x=2y26y+3 9.6, 9.7 Recognizing Conic Sections Example: Follow the example on p. 493 Find the equation of the parabola whose focus is (-3,1) and whose directrix is y=-4. Dealing with general form quadratic equations with xy terms is a bit more complex. We will not plot equations with xy terms, but when you encounter one you need to recognize what type of conic section it is. Example: Which conic section will the following equation be? x 2 + 4 xy + y 2 −10x − 8y + 16 = 0 9.8 Systems of Quadratics Review the three methods for solving systems of linear equations. (graphs, elimination, and linear combination) 1. What methods could you use to solve a system of 3 equations and 2 unknowns in which one or both of the equations is a quadratic polynomial? 4 x 2 + y 2 = 104 Example 1. Solve the system 2x 2 − 3y 2 = 38 3x 2 − 4 y 2 + 7y = 33 Example 2 Solve the system 12x 2 + 4 y 2 = 228 Example 3: Check the answer to example 2 by graphing x 2 + y 2 − 2x + 3y = 8 Example 4. Solve the system 2x + 4 y = −4