Answer
See the graph
![](https://gradesaver.s3.amazonaws.com/uploads/solution/c8036576-c001-44f8-8d17-4f7f0d33ff4a/result_image/1595161373.png?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T013852Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=8aacf57e6bbfd78338a34b851e92873799746a65c46870b26541058bed1592d9)
Work Step by Step
We are given:
$y=-\frac{1}{2}x^2+4x+1$
Let's list some values:
$x=-2 \rightarrow y=7$
$x=-1 \rightarrow y=\frac{9}{2}$
$x=0 \rightarrow y=1$
$x=1 \rightarrow y=\frac{9}{2}$
$x=2 \rightarrow y=7$
The x-coordinate of the vertex is given by $x=\frac{-b}{2a}=\frac{-4}{2.\frac{-1}{2}}=4$
Find the y-coordinate of the vertex: $y=-\frac{1}{2}(4)^2+4(4)+1=9$
Hence, the vertex is $(4,9)$.