Question 10.1.3: Finding the vertex, focus, and directrix Find the vertex, fo...

First use x = -b/(2a) to find the x-coordinate of the vertex:

To find the y-coordinate of the vertex, let x = -3/2 in y=2 x^{2}+6 x=7:

The vertex is (-3/2, -23/2). Since a = 2, the parabola opens upward. Use 2 = 1/(4p) to get p = 1/8. Since the parabola opens upward, the directrix is 1/8 unit below the vertex and the focus is 1/8 unit above the vertex. The directrix is y = -93/8, and the focus is (-3/2, -91/8).

You can graph y_{1}=-93 / 8 and y_{2}=2 x^{2}+6 x-7, as shown in Fig. 10.7, to check the position of the parabola and its directrix.