What are the vertex, focus, and directrix of the parabola with equation y = x^2 -6x + 15?
To find the vertex of the parabola, we need to complete the square:
y = x^2 - 6x + 15
y = (x^2 - 6x + 9) - 9 + 15 (adding and subtracting 9 inside the parentheses)
y = (x - 3)^2 + 6
So the vertex of the parabola is (3,6).
To find the focus and directrix, we need to use the fact that the parabola is of the form y = 1/(4p)(x - h)^2 + k, where (h,k) is the vertex and p is the distance from the vertex to the focus and from the vertex to the directrix.
In our case, h = 3 and k = 6, so the equation of the parabola becomes:
y = 1/4p(x - 3)^2 + 6
Comparing this to the standard form, we can see that 1/4p = 1, so p = 1/4.
Therefore, the focus is located at a distance of 1/4 units above the vertex, which is (3,6+1/4) = (3, 25/4). The directrix is located at a distance of 1/4 units below the vertex, so it is the horizontal line y = 23/4.
Therefore, the vertex is (3, 6), the focus is (3, 25/4), and the directrix is y = 23/4.