Question

what are the vertex, focus, and directrix of the parabola with equation y=x^2-10x+33?

Answer 1

yikes
vertex is easiest

Answer 2

first coordinate of the vertex is \(-\frac{b}{2a}\) which, in your case, is \(-\frac{-10}{2}=5\)

Answer 3

second coordinate is what you get when you replace \(x\) by \(5\)
i get \(8\) so the vertex is \((5,8)\)

Answer 4

yeah i got the vertex but what's the focus

Answer 5

that means you can write it as
\[y=(x-5)^2+8\] or
\[y-8=(x-5)^2\]

Answer 6

this looks like \(4p(y-k)=(x-h)^2\) (kinda) where \(4p=1\) so \(p=\frac{1}{4}\)

Answer 7

that makes the focus \(\frac{1}{4}\) unit above the vertex
since the vertex is \((5,8)\) the focus is \((5,8+\frac{1}{4})\) or
\[(5,\frac{33}{4})\]

Answer 8

if (god help me) that is not one of your "answer choices" it could be
\[(5,8\tfrac{1}{4})\]

Answer 9

you got it?
the directrix is \(\frac{1}{4}\) units down at \(y=\frac{31}{4}\)

Answer 10

so. what you're saying is that
vertex (5,8)
focus (5, 8.25)
directrix (y=7.75

Answer 11

lol
i thought i had the bases covered with fractions, and mixed numbers
but now decimals
yes, that is correct

Answer 12

xD yeesss. thank you, my savior.

Answer 13

yw!