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.
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bobpursley
it did. thanks so much!
Thanks Bob! Just happy to help. :)
You're welcome! If you have any more questions, feel free to ask.
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