What are the coordinates of the focus of the parabola?
y=1/8x^2+3x
To find the focus of the parabola given by the equation y=1/8x^2+3x, we need to first write the equation in standard form:
y = 1/8x^2 + 3x
y = 1/8(x^2 + 24x)
Now, complete the square to get the equation in standard form:
y = 1/8(x^2 + 24x + 144) - 1/8(144)
y = 1/8(x + 12)^2 - 18
Therefore, the equation is now in standard form: y = a(x-h)^2 + k where (h,k) is the vertex of the parabola.
Comparing with the general equation of a parabola, we know that (h,k) is (-12, -18).
Now, the focus of the parabola is located at the point (-12, -18 + 1/(4a)).
In this case, a = 1/8. Therefore, the focus of the parabola is (-12, -18 + 1/(4a)) = (-12, -18 + 8) = (-12, -10).
Therefore, the coordinates of the focus of the parabola are (-12, -10).