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
either#color(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)#
