Write each of the following equations in vertex form and determine the vertex, focus, directrix, and axis of symmetry of the parabola. Graph the parabola.

Question

x²= –16y + 64

Answer 1

This equation has NO linear term for x (no +6x or -2x, etc.) which means the following:

We don't have to "complete the square" to put this in vertex form

The axis of symmetry is x = 0 (ie the y-axis)

The vertex lies on the y-axis

We don't have much to do to put this in vertex form

x² - 64 = -16y

y = -1/16·x² + 4

or y = -1/16(x - 0)² + 4 Vtx: (0, 4)

1/4p = - 1/16 ; p = -4 (so the focus is 4 units below the vertex, while the directrix is 4 units above)

Focus: (0 , 0) ; Directrix: y = 8

Graph should be concave down, vertex at (0 , 4), with x-ints at (-8 , 0) and (8 , 0) (2p away from the focus).