Answer
The axis of symmetry is $x=7$.
The vertex is $(7,\frac{41}{2})$.
Work Step by Step
The given function is
$h(x)=-\frac{1}{2}x^2+7x-4$
It is in the form of $f(x)=ax^2+bx+c$
We have $a=-\frac{1}{2},b=7$ and $c=-4$
(a)
Equation for the axis of symmetry.
$\Rightarrow x=-\frac{b}{2a}$
Substitute $-\frac{1}{2}$ for $a$ and $7$ for $b$.
$\Rightarrow x=-\frac{7}{2(-\frac{1}{2})}$
Simplify.
$\Rightarrow x=-\frac{7}{-1}$
$\Rightarrow x=7$
The axis of symmetry is $x=7$.
(b)
Substitute $7$ for $x$ in the given function.
$\Rightarrow h(7)=-\frac{1}{2}(7)^2+7(7)-4$
Simplify.
$\Rightarrow h(7)=-\frac{49}{2}+49-4$
$\Rightarrow h(7)=-\frac{49}{2}+45$
$\Rightarrow h(7)=\frac{-49+90}{2}$
$\Rightarrow h(7)=\frac{41}{2}$
The vertex is $(7,\frac{41}{2})$.
