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write the equations of the parabola, the directrix, and the axis of symmetry.
vertex: (-4,2)
focus: (-4,6)
if someone could explain how to do this problem then that would be great! thanks in advance!
Formats to help you find the equation for a parabola:
(x - h)^2 = 4p(y - k)
Vertex = (h, k)
Focus = (h, k + p)
Directrix: y = k - p
You are given the vertex (-4,2) and the focus (-4,6).
Since we know k, which is 2, we can figure out p. Format = (h, k + p) for focus. Therefore, p = 4.
I'll set this up and let you take it from there:
[x - (-4)]^2 = 4(4)(y - 2)
The axis of symmetry is: x = -4 (vertex x-value).
I hope this will help.
good post. mathguru, if you would like to join us here at jiskha as a volunteer teacher, email me at
bobpursley
it did. thanks so much!
Thanks Bob! Just happy to help. :)
Anonymous
answered8 years ago
Step-by-Step Bot
answered1 year ago
You're welcome! If you have any more questions, feel free to ask.
Explain Bot
answered11 months ago
To find the equations of the parabola, the directrix, and the axis of symmetry, follow these steps:
1. Use the given information to determine the vertex and focus of the parabola. In this case, the vertex is (-4,2) and the focus is also (-4,6).
2. Now, recall the standard form of a parabola: (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus (or vice versa).
3. Substitute the vertex coordinates into the equation to find h and k. In this case, h = -4 and k = 2.
4. Determine the value of p by using the fact that the focus is (h, k + p). In this case, since the focus is (-4,6), we can deduce that k + p = 6. Plugging in the value of k, we get 2 + p = 6, which gives p = 4.
5. Replace the values of h, k, and p into the standard form equation. This gives us (x - (-4))^2 = 4(4)(y - 2), which simplifies to (x + 4)^2 = 16(y - 2).
The equation of the parabola is therefore (x + 4)^2 = 16(y - 2).
6. The axis of symmetry is a vertical line passing through the vertex. In this case, the vertex is (-4,2), so the equation of the axis of symmetry is x = -4.
7. To find the equation of the directrix, recall that it is a horizontal line given by the equation y = k - p. In this case, the vertex is (-4,2) and p = 4, so the equation of the directrix is y = 2 - 4, which simplifies to y = -2.
Therefore, the equations of the parabola, the directrix, and the axis of symmetry are:
Parabola: (x + 4)^2 = 16(y - 2)
Directrix: y = -2
Axis of symmetry: x = -4