Intercepts
The y (also know as f(x)) intercept is the value of y when x=0
Given y=f(color(blue)x)=color(blue)x^2-16color(blue)x+63
when color(blue)x=color(blue)0
color(white)("XXX")y=f(color(blue)0)=color(blue)0^2-16 * color(blue)0+63=color(red)63
So the y intercept is at y=63
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[Depending upon prior knowledge, the following could be greatly reduced; however, I decided to provide the complete detailed expansion].
The x-intercepts are the values of color(blue)x for which f(color(blue)x)=color(blue)0
color(white)("XXX")f(color(blue)x)=color(blue)0=x^2-16x+63
We know that if f(x) can be factored into two binomials of the form:
color(white)("XXX")(x+color(magenta)p)(x+color(lime)q)
then by expansion
color(white)("XXX")f(x)=x^2+(color(magenta)p+color(lime)q)x+color(magenta)pcolor(lime)q
We are given that
color(white)("XXX")f(x)=x^2-16x+63
So if f(x) can be factored into the form (x+color(magenta)p)(x+color(lime)q)
then
color(white)("XXX")color(magenta)pcolor(magenta)q=63
and
color(white)("XXX")color(magenta)p+color(lime)q=-16
That is, we are hoping to find factors of 63 which add up to (-16).
Because color(magenta)pcolor(lime)q=63>0 we know that color(magenta)p and color(lime)q must have the same sign;
Furthermore, since color(magenta)p+color(lime)q=-16, both color(magenta)p and color(lime)q# must be negative.
We can start building a list of such possible factors:
color(white)("XXX"){:
(color(white)("xx")color(magenta)p,color(white)("xx")color(lime)q,color(white)("XXX"),"Sum"),
(-1,-63,,-64),
(-3,-21,,-24),
(-7,-9,,-16)
:}
There is no need to continue beyond this point since we have found a pair of values color(magenta)p and color(lime)q# that meet our requirements.
We now can write
color(white)("XXX")0=(xcolor(magenta)(-7))(xcolor(lime)(-9))
which implies
eithercolor(white)("XXX")x=7 or color(white)("XXX")x=9
That is the x-intercepts are at x=7 and x=9
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Vertex
f(x)=x^2-16x+63
can be converted into vertex form f(x)=(x-color(red)a)^2+color(blue)b with vertex at (color(red)a,color(blue)b), as follows
f(x)=x^2-16xcolor(orange)(+8^2)+63color(orange)(-8^2)
color(white)("XXX")=(x-color(red)8)+color(blue)1
That is, the vertex is at (x,y)=(8,1)
