Answer
Vertex: $\left(2,3\right)$
Axis of Symmetry: $x=2$
Domain: set of all real numbers
Range: $\{y|y\le3\}$
Graph of $f(x)=-x^2+4x-1$
Work Step by Step
To find the properties of the given function, $
f(x)=-x^2+4x-1
,$ convert to the form $f(x)=a(x-h)^2+k$.
Grouping the $x$-variables together and making the coefficient of $x^2$ equal to $1$, the given equation is equivalent to
\begin{align*}
f(x)&=(-x^2+4x)-1
\\
f(x)&=-(x^2-4x)-1
.\end{align*}
Completing the square of the right-side expression by adding $\left(\dfrac{b}{2}\right)^2,$ the equation above is equivalent to
\begin{align*}
f(x)&=-\left(x^2-4x+\left(\dfrac{4}{2}\right)^2\right)+\left[-1-(-1)\left(\dfrac{4}{2}\right)^2\right]
\\\\
f(x)&=-\left(x^2-4x+4\right)+\left[-1+4\right]
\\\\
f(x)&=-\left(x-2\right)^2+3
.\end{align*}(Note that $a\left(\dfrac{b}{2}\right)^2\Rightarrow
(-1)\left(\dfrac{4}{2}\right)^2
$ should be subtracted as well to cancel out the term that was added to complete the square.)
Since the vertex of the function $f(x)=a(x-h)^2+k$ is given by $(h,k)$, then the vertex of the quadratic function, $
f(x)=-\left(x-2\right)^2+3
$, is $
\left(2,3\right)
$.
The axis of symmetry of the function $f(x)=a(x-h)^2+k$ is given by $x=h$. With $h=
2
$ then the axis of symmetry is $
x=2
$.
To graph the parabola, find points that are on the parabola. This can be done by substituting values of $x$ and solving the corresponding value of $y$. Let $y=f(x).$ Then $
y=-x^2+4x-1
$. Substituting values of $x$ and solving $y$ results to
\begin{array}{l|r}
\text{If }x=0: & \text{If }x=1:
\\\\
y=-(0)^2+4(0)-1 & y=-(1)^2+4(1)-1
\\
y=0+0-1 & y=-1+4-1
\\
y=-1 & y=2
.\end{array}
Hence, the points $
(0,-1)
$ and $
(1,2)
$ are on the parabola. Reflecting these points about the axis of symmetry, the points $
(3,2)
$ and $
(4,-1)
$ are also on the parabola.
Using the points $\{
(0,-1), (1,2),
\left(2,3\right),
(3,2), (4,-1)
\}$ the graph of the parabola is determined (see graph above).
Using the graph, the domain (values of $x$ used in the graph) is the set of all real numbers. The range (values of $y$ used in the graph) is $
\{y|y\le3\}
$.
