Question 625231
{{{f(x)=(1/2)(x+9)^2+7 }}} is a quadratic function.
As such, its graph is a parabola.
Unlike linear functions, which graph as straight lines, quadratic functions are  allowed to have no x-intercept.
They may approach the x-axis for a while, but then they change their mind and turn away from the x-axis, never crossing it.
 
For your function: 
{{{f(x)=7}}} for {{{x=-9}}}
For any other value of x,
{{{(x+9)^2>0}}} and {{{f(x)>7}}}
There is no x-intercept, because {{{f(x)>=7}}} for all values of x.
The graph never crosses the x-axis (the y=0 line), because {{{f(x)}}} is never zero.
The function has a minimum at {{{x=-9}}} with {{{f(-9)=7}}}
The vertex of the parabola is at that point: point(-9,7).
Your function graphs as this:
{{{graph(300,300,-18,2,-5,45,(x+9)^2/2+7)}}}
 
Quadratic functions can be written in the form
{{{f(x)=a(x-h)^2+k}}} or {{{y=a(x-h)^2+k}}}
with {{{a}}}, {{{h}}} , and {{{k}}} being some real numbers (and a must not be zero).
When written in that form, (h,k) is the vertex, and {{{a}}} tells you how curvy the curve is and which way it goes.
The larger the absolute value of {{{a}}}, the curvier the curve is.
If {{{a}}} is positive, the curve goes up from the vertex on both sides, as {{{(x-h)^2}}} gets larger and larger. It looks like a smiley mouth.
If {{{a}}} is negative, the curve goes down from the vertex on both sides, as {{{(x-h)^2}}} gets larger and larger. It's frowning.
Your curve is smiling at you.