Find the equation of the parabola with the following properties. Express your answer in standard form

Question

Find the equation of the parabola with the following properties. Express your answer in standard form

Focus at (-3,-2)

Directrix is the x-axis

Answer 1

y= (-1/4)(x+3)^2 -1

vertex is half way between the focus and directrix

It's a downward opening parabola, general form

y= a(x-h)^2 + k where (h,k) = vertex= (-3,-1)

plug in another point on the parabola to solve for a which gives

am answer with either x coefficient = -1'/4 or =4 Check the math.

one or the other is right

another point is the y intercept = 9a-1

another point is directly to the right of the focus (-1, -2) It's 2 down from the directrix and 2 to the right of the focus, equidistant. plug that point into y= a(x+3)^2 -1 and solve for "a"

-2 = a((-1+3)^2 -1

-2 = 4a -1

4a = -1

a = -1/4

the parabola is

y = (-1/4)(x+3)^2 -1

Answer 2

Focus (- 3, - 2) and directrix y = 0

Equation of parabola y - k = 4p(x - h)²_.

Vertex (h, k) = (- 3, - 1)

p = (- 2 - 0))/2 = - 1.

Equation of parabola: y - (-1) = - 4(x -(- 3))²; y = - 4(x + 3)² - 1.