Given:
[tex]y=-x^2-2x+3[/tex]a) To find the vertex:
Here, a=-1, b=-2, and c=3
We know that the formula to find the x- coordinate of the vertex is given by,
[tex]\begin{gathered} -\frac{b}{2a}=-\frac{(-2)}{2(-1)} \\ =-1 \end{gathered}[/tex]Substitute x=-1 in the given equation we get,
[tex]\begin{gathered} y=-(-1)^2-2(-1)+3 \\ =-1+2+3 \\ =4 \end{gathered}[/tex]Hence, the vertex of the graph is (-1, 4).
b) To find the range of the graph:
Let us find the y-intercept.
Put x=0, we get
[tex]\begin{gathered} y=-(0)^2-2(0)+3 \\ =3 \end{gathered}[/tex]From the figure, we observe that
The range of the graph is
[tex]\lbrack0,4\rbrack[/tex]c) To find the domain of the graph:
Let us find the x-intercept.
Put y=0, we get
[tex]\begin{gathered} -x^2-2x+3=0 \\ (x+3)(x-1)=0 \\ x=-3,1 \end{gathered}[/tex]From the figure, we observe that,
The domain of the graph is,
[tex]\lbrack-3,0)[/tex]