Question

Using a directrix of y = −2 and a focus of (1, 6), what quadratic function is created? Question 3 options:

Answer 1

If your directrix is a "y=" line, that means that the parabola opens either upwards or downwards (as opposed to the left or the right). Because it is in the character of a parabola to "hug" the focus, our parabola opens upwards. The vertex of a parabola sits exactly halfway between the directrix and the focus. Since our directrix is at y = -2 and the focus is at (1, 6) AND the parabola opens upward, the vertex is going to sit on the main transversal, which is also the "line" the focus sits on. The focus is on the line x = 1, so the vertex will also have that x coordinate. Halfway between the y points of the directrix and the focus, -2 and 6, respectively, is the y value of 2. So the vertex sits at (1, 2). The formula for this type of parabola is $(x-h)^2=4p(y-k)$ where h and k are the coordinates of the vertex and p is the DISTANCE that the focus is from the vertex. Our focus is 4 units from the vertex, so p = 4. Filling in our h, k, and p: $(x-1)^2=4(4)(y-2)$. Simplifying a bit gives us $(x-1)^2=16(y-2)$. We can begin to isolate the y by dividing both sides by 16 to get $\frac{1}{16}(x-1)^2=y-2$. Then we can add 2 to both sides to get the final equation $f(x)=\frac{1}{16}(x-1)^2+2$, choice 4 from above.