Answer:
The equation of a parabola is
[tex]x = \frac{1}{4(f - h)} (y - k) ^{2} + h[/tex]
Step-by-step explanation:
(h,k) is the vertex and (f,k) is the focus.
Thus, f = 1, k = −4.
The distance from the focus to the vertex is equal to the distance from the vertex to the directrix: f - h = h - 2.
Solving the system, we get h = 3/2, k = -4, f = 1.
The standard form is:
[tex]x = - \frac{y ^{2} }{2} - 4y - \frac{13}{2} [/tex]
The general form is:
[tex]2x + {y}^{2} + 8y + 13 = 0[/tex]
The vertex form is:
[tex]x = - \frac{(y + 4) ^{2} }{2} + \frac{3}{2} [/tex]
The axis of symmetry is the line perpendicular to the directrix that passes through the vertex and the focus: y = -4.
The focal length is the distance between the focus and the vertex: 1/2.
The focal parameter is the distance between the focus and the directrix: 1.
The latus rectum is parallel to the directrix and passes through the focus: x = 1.
The length of the latus rectum is four times the distance between the vertex and the focus: 2.
The eccentricity of a parabola is always 1.
The x-intercepts can be found by setting y = 0 in the equation and solving for x.
x-intercept:
[tex]( - \frac{13}{2} \: ,0)[/tex]
The y-intercepts can be found by setting x = 0 in the equation and solving for y.
y-intercepts:
[tex](0, - 4 - \sqrt{3)} [/tex]
[tex](0, - 4 + \sqrt{3)} [/tex]
