he y-coordinate of the vertex, k, we can either use the formula (4ac-b²)/(4a) in Figure B or plug the x-coordinate we found into y=ax²+bx+c. We’ll do the latter, using x=3 we just found:
y = ax² + bx + c = 1*(3)² – 6 (3) + 15 = 9 – 18 + 15 = 6.
Thus, the vertex is (3,6).
TEST TAKING TIP #1: Notice here that we have only one answer that has vertex (3,6), and that answer is A. Thus, at this point, I would double check my calculations very carefully (as always, but especially in this case, as we would be betting the house on it), and pick that answer. Then move on to the next question from there. I’ll do this especially when the test is timed. We’ll continue the rest of the calculations, however, to be sure you know how to do the rest.
How to Find the Focus
The focus lies on the axis of symmetry of the parabola, and has the y-coordinate k+1/(4a). Because we just found the vertex to be (3,6), we know the axis of symmetry to be x=3, and the focus lies on that line. See Figure B. Its x-coordinate is therefore x=3. Because in our problem a = 1, the y-coordinate of the focus is 1/(4a) = 1/(4*1) = 1/4 north of the vertex. Thus, y = 6 + 1/4 = 6.25, and our focus is (3,6.25), which is what we were expecting from our selection of multiple choice answer A.
TEST TAKING TIP #2: Note that we know that the vertex x-coordinate and focus x-coordinate are always the same. So even before we made any calculations in this problem, once we knew the parabola opened upward, we could rule out multiple choice answer D, as that option presents a vertex and focus with different x-coordinates.
How to Find the Directrix
Finally, we can find the directrix of a parabola by noting that it will be a horizontal line and south of the vertex of the upward opening parabola, as we said above. Once again, see Figure B. Once you know the y=coordinate of the vertex, k, it is given by y = k – p, where p = 1/(4a). Thus, as we calculated for the focus, above:
p = 1/(4a) = 1/(4*1) = 1/4
and the directrix is y = 6 – 1/4 = 5.75, once again confirming our answer of multiple choice option A.
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