SECTION 10.2 The Parabola 683 Recall that a is the distance from the vertex to the focus of a parabola. When graphing the parabola = y ax 4 , 2 it is helpful to determine the “opening” by finding the points that lie directly above and below the focus ( ) a, 0 . Do this by substituting = x a in = y ax 4 , 2 so = ⋅ = y a a a 4 4 , 2 2 or = ± y a2 . The line segment joining the two points ( ) a a , 2 and ( ) − a a , 2 is called the latus rectum ; its length is a4 . THEOREM Equation of a Parabola: Vertex at 0, 0 , ( ) Focus at a, 0 , ( ) > a 0 The equation of a parabola with vertex at ( ) 0, 0 , focus at ( ) a, 0 , and directrix = − > x a a , 0, is = y ax 4 2 (2) Figure 5 = y x 12 2 F 5 (3, 0) D: x 5 23 x y V (3, 6) Latus rectum (0, 0) (3, 26) 6 26 6 26 Finding the Equation of a Parabola and Graphing It Find an equation of the parabola with vertex at ( ) 0, 0 and focus at ( ) 3, 0 . Graph the equation. Solution EXAMPLE 1 The distance from the vertex ( ) 0, 0 to the focus ( ) 3, 0 is = a 3. Then, the equation of the parabola is = = y ax y x 4 12 2 2 To graph the parabola, find the two points that determine the latus rectum by substituting = x 3. Then = = ⋅ = = ± y x y 12 12 3 36 6 2 The points ( ) 3, 6 and ( ) − 3, 6 determine the latus rectum. These points help graph the parabola because they determine the “opening.” See Figure 5. Equation (2) a 3 = x 3 = Solve for y. Now Work PROBLEM 23 Graphing a Parabola Using a Graphing Utility Graph the parabola = y x 12 . 2 Solution EXAMPLE 2 To graph the parabola y x 12 2 = on a TI-84 Plus CE graphing calculator, we need to graph the two functions = Y x 12 1 and = − Y x 12 2 , since = y x 12 2 does not define y as a function of x. See Figure 6(a). Figure 6(b) shows the graph using Desmos, which can graph the equation in implicit form. Figure 6 y x 12 2 = 20 Y1 5 12x 220 214 50 Y2 5 2 12x (a) (b) By reversing the steps used to obtain equation (2), it follows that the graph of an equation of the form = y ax 4 2 is a parabola; its vertex is at ( ) 0, 0 , its focus is at ( ) a, 0 , its directrix is the line = − x a, and its axis of symmetry is the x -axis.
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