Final answer:
To find the vertex and focus of a parabola, the quadratic equation must be put into standard form. Completing the square in the given equation does not result in coordinates that match any of the provided options, suggesting an error in the question or options. The correct vertex is (-1/4, -33/16), and the focus is (-1/4, -1/16).
Step-by-step explanation:
The boundary of the lawn in front of a building is specified by the parabolic equation x^2/16 + x - 2. To find the coordinates of the focus (h, k) and the vertex (a, b) of the parabola, we first need to express the equation in the standard form of a parabola, which is (x - h)^2 = 4p(y - k) for a parabola that opens upwards or downwards, and (y - k)^2 = 4p(x - h) for a parabola that opens to the left or right.
We notice that the given equation is not in the standard form, so we would complete the square to express it properly:
x^2/16 + x = y + 2
(x^2/16 + x + 1/16) - 1/16 = y + 2
(x + 1/4)^2 = 16(y + 33/16)
Now in the form (x - h)^2 = 4p(y - k), we can see that h = -1/4, k = -33/16, and 4p = 16, so p = 4.
Therefore, the vertex is at the point (h, k), or more accurately, (-1/4, -33/16). The focus lies a distance 'p' above the vertex (for an upwards opening parabola), so the focus is at the point (h, k + p) which is (-1/4, -1/16).
None of the options provided in the question correctly represent the focus and vertex for the given parabola. There might be a mistake in either the formulation of the question or the provided options. It is essential to write the correct equation of the parabola before identifying the focus and vertex.