684 CHAPTER 10 Analytic Geometry For the remainder of this section, the direction “Analyze the equation” means to find the vertex, focus, and directrix of the parabola and graph it. Analyzing the Equation of a Parabola Analyze the equation = y x8 . 2 Solution EXAMPLE 3 The equation = y x8 2 is of the form = y ax 4 , 2 where = a4 8, so = a 2. Consequently, the graph of the equation is a parabola with vertex at ( ) 0, 0 and focus on the positive x-axis at ( ) ( ) = a, 0 2, 0 . The directrix is the vertical line = a 2 units to the left of the y-axis. That is, = − x 2. The two points that determine the latus rectum are obtained by substituting = x 2 in the equation = y x8 . 2 Then = y 16, 2 so = ± y 4. The points ( ) − 2, 4 and ( ) 2, 4 determine the latus rectum. See Figure 7(a) for the graph drawn by hand. Figure 7(b) shows the graph obtained using GeoGebra. Figure 7 = y x8 2 x y 2 5 5 25 V (2, 2 (2, 4) (0, 0) (a) (b) 5 4) D: x 5 22 F 5 (2, 0) Latus rectum TIP The axis of symmetry corresponds to the variable that appears to the power 1. When the coefficient of that variable is positive, the parabola opens right or up. When the coefficient is negative, the parabola opens left or down. j Recall that we obtained equation (2) after placing the focus on the positive x-axis. Placing the focus on the negative x-axis, positive y-axis, or negative y-axis results in a different form of the equation for the parabola. The four forms of the equation of a parabola with vertex at ( ) 0, 0 and focus on a coordinate axis a distance a from ( ) 0, 0 are given in Table 1, and their graphs are given in Figure 8. Notice that each graph is symmetric with respect to its axis of symmetry. Equations of a Parabola: Vertex at 0, 0 ; ( ) Focus on an Axis; 0 a > Vertex Focus Directrix Equation Description 0, 0 ( ) , 0 a( ) x a = − y x 4 2 a = Axis of symmetry is the x-axis, the parabola opens right 0, 0 ( ) , 0 a ( ) − x a = y x 4 2 a = − Axis of symmetry is the x-axis, the parabola opens left 0, 0 ( ) 0, a ( ) y a = − x y 4 2 a = Axis of symmetry is the y-axis, the parabola opens up (is concave up) 0, 0 ( ) 0, a ( ) − y a = x y 4 2 a = − Axis of symmetry is the y-axis, the parabola opens down (is concave down) Table 1 Figure 8 x (d) x2 5 24ay V x y (c) x2 5 4ay V y (b) y2 5 24ax V x y V (a) y2 5 4ax x y F 5 (0, 2a) D: y 5 a F 5 (0, a) D: y 5 2a D: x 5 a F 5 (2a, 0) D: x 5 2a F 5 (a, 0)
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