SECTION 10.2 The Parabola 689 and its focus is at ( )a 0, . Since ( ) 4, 3 is a point on the graph, this gives = ⋅ = a a 4 4 3 4 3 2 The receiver should be located 4 3 feet (1 foot, 4 inches) from the base of the dish, along its axis of symmetry. x ay x y 4 ; 4, 3 2 = = = Solve for a. Now Work PROBLEM 71 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 10.2 Assess Your Understanding 1. The formula for the distance d from P x y , 1 1 1 ( ) = to P x y , 2 2 2 ( ) = is d = . (p. 14 ) 2. To complete the square of x x4 , 2 − add . (p. A29) 3. Use the Square Root Method to find the real solutions of x 4 9. 2 ( ) + = (p. A49) 4. The point that is symmetric with respect to the x -axis to the point 2, 5 ( ) − is . (pp. 21–23) 5. To graph y x 3 1, 2 ( ) = − + shift the graph of y x2 = to the right units and then 1 unit. (pp. 112–120) 6. The graph of y x 3 5 2 ( ) = − − has vertex and axis of symmetry . (pp. 157–166) Concepts and Vocabulary 7. A(n) is the collection of all points in a plane that are the same distance from a fixed point as they are from a fixed line. The line through the focus and perpendicular to the directrix is called the of the parabola. 8. For the parabola y ax 4 , 2 = the line segment joining the two points a a , 2 ( ) and a a , 2 ( ) − is called the . 9. Interactive Figure Exercise Exploring the Graph of a Parabola Open the “Parabola Left_Right” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Grab the directrix slider so that the directrix is x 2. = − Grab the point F (the focus) and move to 4, 2 . ( ) Check the box “Show Point on Parabola”. Grab point P and move it around the Cartesian Plane. Notice the distance from P to F and the distance from P to the directrix are computed. Move P to the point 4, 8 . ( ) What is the distance from P to F? (b) Continue to trace out the parabola. Notice the focus is to the right of the directrix.The parabola opens (up/ down/left/right). (c) What is the distance, a, from the focus to the vertex? What is the distance, a, from the vertex to the directrix? (d) What are the coordinates of the vertex? Write your answer as an ordered pair. (e) Check the box “Show Parabola”. What is the equation of the parabola? Write your answer in the form y k a x h 4 . 2 ( ) ( ) − = − (f) Reset the interactive figure by clicking . Move the focus (point F) to 8, 2 . ( ) − Change the directrix to x 2. = What are the coordinates of the vertex? (g) Check the box “Show Point on Parabola”. Grab point P and move it around the Cartesian Plane. Notice the distance from P to F and the distance from P to the directrix is computed. Move P to the point 8, 8. ( ) − − What is the distance from P to F? (h) Continue to trace out the parabola. Notice the focus is to the left of the directrix. The parabola opens (up/down/left/right). (i) Check the box “Show Parabola”. What is the equation of the parabola? Write your answer in the form y k a x h 4 . 2 ( ) ( ) − = − 10. Interactive Figure Exercise Exploring the Graph of a Parabola Open the “Parabola Up_Down” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Use the sliders to set a to h 2; − to 0; k to 0. What are the coordinates of the vertex? What are the coordinates of the focus? What is the distance from the vertex to focus? (b) Does the parabola open up or down? (c) Change a to 2, but leave h and k set to 0. What are the coordinates of the vertex? What are the coordinates of the focus? What is the distance from the vertex to focus? (d) Does the parabola open up or down? (e) Leave a set to 2. Change h to 1 and k to 3. What is the distance from the vertex to focus? What are the coordinates of the vertex? What is the equation of the parabola? What is the equation of the directrix? 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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