Find the vertex, focus, and directrix of the parabola and sketch its graph.?

Question

Find the vertex, focus, and directrix of the parabola and sketch its graph:
y + 12x - 2x^2 = 16

Answer 1

Find the vertex, focus, and directrix of the parabola and sketch its graph.

Put the equation of the parabola in vertex form.

y + 12x - 2x² = 16
y = 2x² - 12x + 16
y - 16 = 2(x² - 6x)
y - 16 + 2*9 = 2(x² - 6x + 9)
y + 2 = 2(x - 3)²
(1/2)(y + 2) = (x - 3)²

The equation of a vertical parabola with vertex (h, k) is:

4p(y - k) = (x - h)²

4p = 1/2
p = 1/8

This particular parabola is vertical and opens upwards.

The vertex (h, k) = (3, -2).

Since the parabola is vertical the line of symmetry is also vertical and passes thru the vertex. Its equation is:

x = h
x = 3

The focus is also on the line of symmetry at a directed distance of p from the vertex. Its coordinates are:

(h, k + p) = (3, -2 + 1/8) = (3, -15/8)

The directrix is a horizonal line at a directed distance of -p from the vertex. Its equation is:

y = k - p = -2 - 1/8 = -17/8

You'll have to sketch the graph on your own.

Answer 2

X^2-4x-8y+4=0