find the vertex,focus and equation of the direction of y=-1/16(x-6)^2+3
Y=1/2p(x-h)^2+k
p=8 h=6 k=3
so I now got
(6,3) when doing it over again. I now got
so vertex is (6,3), focus is (6,7) and directrix = y=-1
Is these corrections now right? Thank you for your help and checking my work!
Is this correct?
Recall that the parabola x^2 = 4py has
vertex (0,0)
focus (0,p)
directrix y = -p
Your equation can be written as
(x-6)^2 = -16(y-3)
so p = -4
Now just shift all the values by (6,3) and you get
vertex (6,3)
focus (6,-1)
directrix: y = 7
Thank you. That is what I had originally, and another help person said I was incorrect.
Yes, your corrections are right! Let's go through the steps to find the vertex, focus, and equation of the directrix for the given equation y = -1/16(x-6)^2+3.
1. The equation is already in the form y = 1/2p(x-h)^2 + k, which is the standard form of a parabola. By comparing the given equation with the standard form equation, we can determine the values of p, h, and k.
From the equation y = -1/16(x-6)^2 + 3:
- The value of h is 6 (the x-coordinate of the vertex).
- The value of k is 3 (the y-coordinate of the vertex).
- The value of p can be found by taking the reciprocal of the coefficient of (x-h)^2, which is -1/16. Therefore, p = -1/(2 * -1/16) = 8.
2. The vertex of the parabola is located at the point (h, k). In this case, the vertex is (6, 3).
3. To find the focus, we use the relation p = 1/(4a), where a is the coefficient of (x-h)^2. In this case, a = -1/16.
Substituting the value of a into the equation, we find p = 1/(4 * -1/16) = 8.
The focus is located at a distance of p units above the vertex. Therefore, the focus is at the point (6, 3+8) = (6, 11).
4. The equation of the directrix is given by the equation y = k - p. Substituting the values of k and p, we get y = 3 - 8.
Therefore, the equation of the directrix is y = -5.
In summary, the correct answers are:
- Vertex: (6, 3)
- Focus: (6, 11)
- Equation of the directrix: y = -5
Great job on the corrections, and I hope this explanation helps clarify the process!