How do you find the standard equation for a parobala when you do not have the directrix? I do have the focus, but the thing is the focus is not a 0,p value, it is -2,5, so I don't see how it could really be a focus. I'm sorry I can't get the graph in this message, but an explanation on finding the standard equation in general will do. Thanks!!!
<< ...but the thing is the focus is not a 0,p value, it is -2,5...>>
Is the focus (0,-2.5) or (-2.5,0)
you are also missing some more information.
There has to be at least another point given on the parabola.
Did they perhaps give you the vertex?
Here is a nice webpage that explains a lot about the parabola and its variations of equations to express it.
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_conics_directrix.xml
To find the standard equation for a parabola when you have the focus and do not have the directrix, you can use the vertex form of the equation. The vertex form of a parabola is given by:
(y - k) = a(x - h)^2
In this form, the vertex of the parabola is (h, k), and 'a' determines the shape of the parabola. To find the standard equation, we need to determine the values of 'h', 'k', and 'a'.
In your case, you have the focus point (-2, 5), and you want to find the standard equation. However, you mentioned that the focus is not in the form (0, p) as expected. To adjust for this, we need to shift the parabola horizontally so that the new focus becomes (0, p), where p is the distance from the focus to the directrix.
Let's start by shifting the parabola horizontally. To do this, we can substitute (x - h) with (x - h') in the vertex form, where h' is the new x-coordinate of the vertex (which will be the opposite of the x-coordinate of the focus):
(y - k) = a(x - h')^2
(y - k) = a(x + 2)^2
Now, we can see that the vertex is at (-2, k), and we need to determine the value of 'k'. Since the focus is (h', p), we know that the vertex is the midpoint between the focus and the directrix. Therefore, the y-coordinate of the vertex is the average of the y-coordinates of the focus and directrix:
k = (5 + p) / 2
To find the value of 'a', we need to use the relationship between 'a' and the distance from the focus to the vertex. The distance from the focus to the vertex is equal to the absolute value of '1 / (4a)'.
In your question, you did not mention the distance from the focus to the directrix (p). Once you determine the value of 'p', you can calculate 'a' using the distance relationship mentioned above. Then, substitute the values of 'a', 'h' (which is -2), and 'k' into the vertex form equation to obtain the standard equation of the parabola.