Question 1157025
 graphing the following and listing the directions,vertex,axis of symmetry,table with {{{5}}} points and the graph for this equation

{{{f(x)=(x-6)^2+2}}}

from given formula you see that {{{h=6}}} and {{{k=2}}}, so vertex is at ({{{6}}},{{{2}}})

so, we have one point for a graph

For a parabola in standard form:{{{y=ax^2+bx+c}}}
the axis of symmetry is the {{{vertical}}} line that goes through the vertex
{{{x= -b/2a}}}

in your case, {{{x}}} coordinate of the vertex is {{{6}}}=:the {{{vertical}}} line that goes through the vertex
{{{x= 6}}}


directrix:

{{{4p (y-k )= (x-h )^2}}} is the standard equation for an up-down facing parabola with vertex at ({{{h}}}, {{{k }}}) and  a focal length{{{ abs(p)}}}

Rewrite {{{y= (x-6 )^2+2}}} in the standard form:
  
{{{4p(y-2 )= (x-6 )^2}}}.....since {{{4p=1}}}=>{{{p=1/4}}}

{{{y=2-p}}}
{{{y=2-1/4}}}
{{{y=8/4-1/4}}}
{{{y=7/4}}}

table:

{{{x}}}|{{{y}}}
{{{6}}}|{{{2}}}
{{{5}}}|{{{3}}}
{{{4}}}|{{{6}}}
{{{7}}}|{{{3}}}
{{{8}}}|{{{6}}}

{{{drawing( 600, 600, -10, 10, -10, 10, 
circle(6,2,.12), locate(6,2,v(6,2)), 
circle(5,3,.12), locate(5,3,p(5,3)), 
circle(4,6,.12), locate(4,6,p(4,6)),
 circle(7,3,.12), locate(7,3,p(7,3)), 
circle(8,6,.12), locate(8,6,p(8,6)), 
green(line(6,10,6,-10)),
 graph( 600, 600, -10, 10, -10, 10, (x-6)^2+2)) }}}


2.
{{{F(x) = x^2-2x-5}}}.....write in vertex form

{{{F(x) = (x^2-2x+b^2)-b^2-5}}}

{{{F(x) = (x^2-2x+1^2)-1^2-5}}}
{{{F(x) = (x-1)^2-6}}}

vertex | ({{{1}}}, {{{-6}}})

compare to{{{4p (y-k )= (x-h )^2}}}=>{{{4p=1}}}=>{{{p=1/4}}}
{{{y=-6-p}}}
{{{y=-6-1/4}}}
{{{y=-25/4}}}
directrix: {{{y = -25/4}}}

axis of symetry:the {{{vertical}}} line that goes through the vertex
{{{x= 1}}}

table:

{{{x}}}|{{{y}}}
{{{1}}}|{{{-6}}}
{{{2}}}|{{{-5}}}
{{{3}}}|{{{-2}}}
{{{4}}}|{{{3}}}
{{{0}}}|{{{-5}}}

{{{drawing( 600, 600, -10, 10, -10, 10, 
circle(1,-6,.12), locate(1,-6,v(1,-6)), 
circle(2,-5,.12), locate(2,-5,p(2,-5)), 
circle(3,-2,.12), locate(3,-2,p(3,-2)),
 circle(4,3,.12), locate(4,3,p(4,3)), 
circle(0,-5,.12), locate(0,-5,p(0,-5)), 
green(line(1,10,1,-10)),
 graph( 600, 600, -10, 10, -10, 10, (x-1)^2-6)) }}}