Question 236217
<pre><font size = 4 color = "indigo"><b>
{{{f(x)=1/2}}}{{{(x+6)^2+3}}}

The equation

{{{f(x)=a(x-h)^2+k}}}  

is 

a parabola with a vertex at (h,k) and which passes
through the points (h+1,k+a) and (h-1,k+a), and its
axis of symmetry is the vertical line whose equation
is {{{x=h}}}

So in the problem:

{{{f(x)=1/2}}}{{{(x+6)^2+3}}}

{{{a=1/2}}}, {{{h=-6}}}, {{{k=3}}}

[Notice that the sign is changed for h but kept for k]

So the vertex is (h,k) or (-6,3),

The parabola passes through (h-1,k+a) and (h+1,k+a), or

(-6-1,3+{{{1/2}}}) or (-7,{{{3&1/2}}}) and

(-6+1,3+{{{1/2}}}) or (-5,{{{3&1/2}}})

Its axis of symmetry is the vertical line whose equation

is {{{x=h}}} or {{{x=-6}}}

We plot the vertex (-6,3)

{{{drawing(400,400,-10,2,-3,9, graph(400,400,-10,2,-3,9),

line(-6+.1,3,-6-.1,3), line(-6,3+.1,-6,3-.1), line(-6+.1,3+.1,-6-.1,3-.1), line(-6+.1,3-.1,-6-.1,3+.1), locate(-6,3,"(6,3)")

  )}}}

We draw the axis of symmetry which is a vertical line 
through the vertex (the green line below:

{{{drawing(400,400,-10,2,-3,9, graph(400,400,-10,2,-3,9),
green(line(-6,-4,-6,10)),
line(-6+.1,3,-6-.1,3), line(-6,3+.1,-6,3-.1), line(-6+.1,3+.1,-6-.1,3-.1), line(-6+.1,3-.1,-6-.1,3+.1), locate(-6,3,"(6,3)")

  )}}}

Then we plot the two points, 
one on each side of the vertex

(-7,{{{3&1/2}}}) and (-5,{{{3&1/2}}})



{{{drawing(400,400,-10,2,-3,9, graph(400,400,-10,2,-3,9),
green(line(-6,-4,-6,10)),
line(-6+.1,3,-6-.1,3), line(-6,3+.1,-6,3-.1), line(-6+.1,3+.1,-6-.1,3-.1), line(-6+.1,3-.1,-6-.1,3+.1), locate(-6,3,"(6,3)"),

line(-7+.1,7/2,-7-.1,7/2), line(-7,7/2+.1,-7,7/2-.1), line(-7+.1,7/2+.1,-7-.1,7/2-.1), line(-7+.1,7/2-.1,-7-.1,7/2+.1), 

line(-5+.1,7/2,-5-.1,7/2), line(-5,7/2+.1,-5,7/2-.1), line(-5+.1,7/2+.1,-5-.1,7/2-.1), line(-5+.1,7/2-.1,-5-.1,7/2+.1)  )}}}

Finally we sketch in the parabola:

{{{drawing(400,400,-10,2,-3,9, graph(400,400,-10,2,-3,9,
(1/2)(x+6)^2+3),
green(line(-6,-4,-6,10)),
line(-6+.1,3,-6-.1,3), line(-6,3+.1,-6,3-.1), line(-6+.1,3+.1,-6-.1,3-.1), line(-6+.1,3-.1,-6-.1,3+.1), locate(-6,3,"(6,3)"),

line(-7+.1,7/2,-7-.1,7/2), line(-7,7/2+.1,-7,7/2-.1), line(-7+.1,7/2+.1,-7-.1,7/2-.1), line(-7+.1,7/2-.1,-7-.1,7/2+.1), 

line(-5+.1,7/2,-5-.1,7/2), line(-5,7/2+.1,-5,7/2-.1), line(-5+.1,7/2+.1,-5-.1,7/2-.1), line(-5+.1,7/2-.1,-5-.1,7/2+.1)  )}}}

Edwin<pre>