Find the vertex, focus, and directrix of the parabola and graph it.

Question

x^2-2x+8y+9=0.

Answer 1

The parabola equation is x^2 - 2x + 8y + 9 = 0.

The general form of parabola equation is y = ax^2 + bx + c.

Write the above equation in general form of parabola.

8y = - x^2 + 2x - 9.

y = 1/8 (- x^2 + 2x - 9)

The standard form of parabola equation is (x - h)^2 = 4p (y - k), where (h, k) = vertex and p = directed distance from vertex to focus.

Write the equation : x^2 - 2x + 8y + 9 = 0 in standard form of parabola.

x^2 - 2x = - 8y - 9.

To change the expression [ x^2 - 2x ] into a perfect square trinomial add (half the x coefficient)² to each side of the expression

 Here x coefficient = 2. so, (half the x coefficient)² = (2/2)²= 1.

x^2 - 2x + 1= - 8y - 9 + 1

(x - 1)^2 = - 8(y + 1)

(x - 1)^2 = 4(-2) [y - (-1)]

Vertex : (h, k) = (1, - 1), p = directed distance from vertex to focus = -2 and Focus : (h, k + p) = (1, -1-2) = (1, - 3).

Make a table; choose some values for x and find the corresponding values for y.

x		(x, y)
-2
-1
0
1
2
3

Use these ordered pairs to graph the equation.

Draw a coordinate plane.

Plot the points.

Draw a smooth curve through these points.

Comments

Information About Parabolas:

Form of Equation	(x - h)^2 = 4p (y - k)	(y - k)^2 = 4p (x - h)
Vertex	(h, k)	(h, k)
Axis of Symmetry	x = h	y = k
Focus	(h, k + p)	(h + p, k)
Directrix	y = k - p	x = h - p
Direction of Opening	upward if a > 0, downward if a < 0	right if a > 0, left if a < 0
Length of Latus Rectum	| 4p | units	| 4p | units

The standard form of parabola x^2 - 2x + 8y + 9 = 0 is (x - 1)^2 = 4(-2) [y - (-1)].

Dirextrix is y = k - p = - 1 - (- 2) = 1.