SECTION 10.2 The Parabola 685 Analyzing the Equation of a Parabola Analyze the equation = − x y 12 . 2 Solution EXAMPLE 4 The equation = − x y 12 2 is of the form = − x ay 4 , 2 with = a 3. Consequently, the graph of the equation is a parabola with vertex at ( ) 0, 0 , focus at ( ) − 0, 3 , and directrix the line = y 3. The parabola opens down (is concave down), and its axis of symmetry is the y-axis. To obtain the points defining the latus rectum, let = − y 3. Then = x 36, 2 so = ± x 6. The points ( ) − − 6, 3 and ( ) − 6, 3 determine the latus rectum. See Figure 9(a) for the graph drawn by hand. Figure 9(b) shows the graph obtained using Desmos. Notice that Desmos does not require that we solve the equation for y to obtain the graph. Figure 9 =− x y 12 2 D: y 5 3 F 5 (0, 23) (6, 23) (26, 23) (0, 0) x y 26 6 6 V (a) (b) Now Work PROBLEM 43 Finding the Equation of a Parabola Find the equation of a parabola with vertex at ( ) 0, 0 if its axis of symmetry is the x-axis and its graph contains the point ( ) − 1 2 , 2 . Find its focus and directrix, and graph the equation. Solution EXAMPLE 6 The vertex is at the origin, the axis of symmetry is the x-axis, and the graph contains a point in the second quadrant, so the parabola opens to the left. From Table 1, note that the form of the equation is = − y ax 4 2 Figure 10 = x y 16 2 x y V 10 (0, 0) (28, 4) (8, 4) 210 10 210 D: y 5 24 F 5 (0, 4) Finding the Equation of a Parabola Find the equation of the parabola with focus at ( ) 0, 4 and directrix the line = − y 4. Graph the equation. EXAMPLE 5 A parabola whose focus is at ( ) 0, 4 and whose directrix is the horizontal line = − y 4 has its vertex at ( ) 0, 0 . (Do you see why? The vertex is midway between the focus and the directrix.) Since the focus is on the positive y-axis at ( ) 0, 4 , the equation of the parabola is of the form = x ay 4 , 2 with = a 4. That is, = = ⋅ = x ay y y 4 4 4 16 2 ↑ a 4 = Substituting = y 4 in the equation = x y 16 2 yields = x 64, 2 so = ± x 8. The points ( ) 8, 4 and ( ) −8, 4 determine the latus rectum. Figure 10 shows the graph of = x y 16 . 2 Check: Verify the graph drawn in Figure 10 using a graphing utility. Solution (continued)
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