Watch two things in these types of problems. First, the geometry of the situation — i.e. the orientation of the parabola and its various parts; and second, make sure your formulas are properly applied, as there are many k’s, p’s and a’s floating around and are a lot to contend with.
The geometry in this case is straight forward — we have a parabola that is opening upward in the plane. This is because we are quadratic in x (as opposed to y — see what to do here when it’s not), and the leading coefficient a is positive. See Figure A.
Because the parabola is opening upward, our focus will have the highest y-coordinate, the vertex will be south of the focus, and finally the directrix will be furthest south of them all.
With that in mind, to actually find the vertex, focus, and directrix, you can either transform your y=x²-6x+15 into vertex form, or you can leave it, and use formulas for these from standard form. The consideration that I make when deciding which to use is whether the standard form is “easily” made into vertex form. When I look at y=x²-6x+15, I do not see a quick way to turn it into vertex form through factoring (meaning we would have to complete the square). Thus, I would keep y=x²-6x+15 in standard form and use formulas from there. To help us with our calculations, we’ve put together a quick reference “cheat sheet” in Figure B.
We’re also solving this problem in the context of multiple choice, which is great because the answer is somewhere already on the paper! It’s even better because sometimes you won’t have to derive the entire answer to rule out certain ones. (See the Test Taking Tips below, for some tips about how to use multiple choice answers to your advantage.)
How to Find the Vertex
First, we’ll work on the vertex. The vertex x-coordinate is given by x=h=-b/(2a). Recall that when in standard form, the convention is to use coefficients a, b, and c as in ax²+bx+c. Thus, a=1, b=-6, and c=15 in our example, and the x-coordinate of the focus is:
x = -b / (2a) = – (-6) / (2*1) = 6/2 = 3.
To find the 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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