Answer
The function has a maximum value of $15$.
Work Step by Step
The given function is
$\Rightarrow y=-x^2+14x-34$
Subtract $15$ from each side.
$\Rightarrow y-15=-x^2+14x-34-15$
Simplify.
$\Rightarrow y-15=-x^2+14x-49$
Factor out $-1$ from the right side.
$\Rightarrow y-15=-1(x^2-14x+49)$
Write the right side as the square of a binomial.
$\Rightarrow y-15=-1(x-7)^2$
Write in vertex form.
$\Rightarrow y=-1(x-7)^2+15$
The vertex is $(7,15)$. Because $a$ is negative $(a=-1)$, the parabola opens down and the $y-$coordinate of the vertex is the maximum value.
Hence, the function has a maximum value of $15$.
