995 10.1 Parabolas If the directrix is the line x = -p and the focus is 1p, 02, a similar procedure leads to the equation of a parabola with a horizontal axis of symmetry. Parabola with Horizontal Axis of Symmetry and Vertex (0, 0) The parabola with focus 1p, 02 and directrix x = -p has the following equation. y2 =4px This parabola has horizontal axis of symmetry y = 0 and opens to the right if p 70 or to the left if p 60. EXAMPLE 3 Graphing Parabolas Give the focus, directrix, vertex, and axis of symmetry for each parabola. Then use this information to graph the parabola. (a) x2 = 8y (b) y2 = -28x SOLUTION (a) The equation x2 = 8y has the form x2 = 4py, with 4p = 8, so p = 2. The x-term is squared, so the parabola is vertical, with focus 10, p2 = 10, 22 and directrix y = -p = -2. The vertex is 10, 02, and the axis of symmetry is the y-axis (that is, x = 0). See Figure 8. (b) The equation y2 = -28x has the form y2 = 4px, with 4p = -28, so p = -7. The parabola is horizontal, with focus 1-7, 02, directrix x = 7, vertex 10, 02, and the x-axis (that is, y = 0) as axis of symmetry. Because p is negative, the graph opens to the left, as shown in Figure 9. S Now Try Exercises 23 and 27. x y –4 4 4 x 2 = 8y y = –2 F(0, 2) V(0, 0) (4, 2) (–4, 2) x = 0 Figure 8 x y –10 2 10 10 –10 y2 = –28x x = 7 F(–7, 0) V(0, 0) (–7, –14) (–7, 14) y = 0 Figure 9 x y x = –p 0 F(p, 0) y2 = 4px, p > 0 Axis y = 0 x y x = –p 0 F(p, 0) y2 = 4px, p < 0 Axis y = 0
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