To find the equation in standard form for the given parabola, we will use the information provided.
- Axis of symmetry at x = -2 implies that the parabola is in the form (x - h)^2 = 4p(y - k), where (h, k) is the vertex of the parabola.
Using the axis of symmetry, we know that the vertex is (-2, k). We can substitute this into our equation to get (x + 2)^2 = 4p(y - k).
- An x-intercept at 4 means that the parabola passes through the point (4, 0). Substituting this point into the equation, we get (4 + 2)^2 = 4p(0 - k).
- A y-intercept at (0, -96) gives us (0 + 2)^2 = 4p(-96 - k).
Now we have two equations:
6^2 = 4p(-k) -> equation 1
2^2 = 4p(-96 - k) -> equation 2
Simplifying equation 1:
36 = -4pk
Simplifying equation 2:
4 = -384p - 4pk
Simplifying equation 1 gives us: -pk = -9
Thus, k = 9/p.
Substituting the value of k in equation 2 gives us:
4 = -384p - 4p(9/p)
4 = -384p - 36
384p = -40
p = -40/384
p = -5/48
Now that we have the value of p, we can substitute it into equation 1 to find k:
-(-5/48)k = -9
5/48 * k = -9
k = -432/5
Substituting the values of p and k into the equation (x + 2)^2 = 4p(y - k) gives us:
(x + 2)^2 = 4(-5/48)(y + 432/5)
Multiplying through by 48:
48(x + 2)^2 = -20(y + 432/5)
Simplifying further:
48(x + 2)^2 + 20(y + 432/5) = 0
Converting to standard form by expanding the equation gives us:
48x^2 + 192x + 192 + 20y + 864 = 0
48x^2 + 192x + 20y + 1056 = 0
Therefore, the equation in standard form for the given parabola is:
48x^2 + 192x + 20y + 1056 = 0.
Therefore, none of the given options (a, b, c, d) matches the equation for the parabola with the given characteristics.