I am supposed to write the standard equation of the parabola with the directrix x=1, and the vertex 6,2. I got y-6=1/4p(x-2)^2. Is this correct?
I am supposed to graph y+3=-1/12(x-1)^2. I tried to find p, and I got 36. I don't think this could be correct, because the graph I am given does not have that high of a range. THe graph almost forms your traditional cross, with just a little bit of space in the positive quadrants, and a lot of space in the negative quadrants. What is the real definition of p? How do you find that? Then what is the focus and the directrix from p? Thanks
I'll answer the first question.
The directrix is vertical so the parabola opens sideways. The equation is therefore of the form:
4p(x - h) = (y - k)^2
The directed distance from the directrix to the vertex is p.
p = 6 - 1 = 5
4p = 4*5 = 20
p is positive so the parabola opens to the right.
The equation of the parabola is therefore:
4p(x - h) = (y - k)^2
20(x - 6) = (y - 2)^2
To determine if your equation for the parabola with the given directrix and vertex is correct, let's first understand the standard equation of a parabola:
The standard equation of a parabola with vertex (h, k), directrix x = a, and focal length p is given by:
(y - k) = 1/(4p)(x - h)^2
Now, let's compare this with your equation:
y - 6 = 1/4p(x - 2)^2
The directrix of the parabola is x = 1, so the value of a is 1. The vertex is given as (6, 2), which means h = 6 and k = 2.
Comparing the two equations, we have:
(y - 6) = 1/(4p)(x - 2)^2
Based on this, we can see that your equation has the correct form, but the values for h and k are swapped. So the correct equation should be:
(y - 2) = 1/(4p)(x - 6)^2
Now, let's move on to the second part of your question regarding the graph with the equation y + 3 = -1/12(x - 1)^2. To find the value of p, we need to rewrite the equation in standard form.
Start by isolating the squared term:
12(y + 3) = -(x - 1)^2
Next, we can rewrite this equation in the standard form:
(y + 3) = -1/12(x - 1)^2
Comparing this equation with the standard form, we have:
(y - k) = 1/(4p)(x - h)^2
From this comparison, we can see that h = 1 and k = -3.
Now, we can determine the value of p by comparing the equation with the standard form:
1/(4p) = -1/12
To solve for p, we can cross-multiply and get:
-12 = 4p
Dividing both sides by 4, we find:
p = -12/4 = -3
So the focal length value, p, is -3.
Now, with the value of p, we can determine the location of the focus and the equation of the directrix.
For a parabola with the equation (y - k) = 1/(4p)(x - h)^2:
- The focus is located at (h, k + 1/(4p)).
- The directrix is a horizontal line given by y = k - 1/(4p).
Plugging in the values h = 1, k = -3, and p = -3, we can calculate:
- The focus is at (1, -3 - 1/(4*(-3))) = (1, -3 + 1/12) = (1, -2.9167) (rounded to four decimal places).
- The directrix is the line y = -3 - 1/(4*(-3)) = -3 - 1/12 = -3.0833 (rounded to four decimal places).
Therefore, the focus is located at (1, -2.9167), and the directrix of the graph is y = -3.0833.