Answer
Vertex: $\left(2,3\right)$
Axis of Symmetry: $x=2$
Domain: set of all real numbers
Range: $\{y|y\le3\}$
Graph of $f(x)=-2x^2+8x-5$
Work Step by Step
To find the properties of the given function, $
f(x)=-2x^2+8x-5
,$ 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)&=(-2x^2+8x)-5
\\
f(x)&=-2(x^2-4x)-5
.\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)&=-2\left(x^2-4x+\left(\dfrac{-4}{2}\right)^2\right)+\left[-5-(-2)\left(\dfrac{-4}{2}\right)^2\right]
\\\\
f(x)&=-2\left(x^2-4x+4\right)+\left[-5+8\right]
\\
f(x)&=-2\left(x-2\right)^2+3
.\end{align*}(Note that $a\left(\dfrac{b}{2}\right)^2\Rightarrow
(-2)\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)=-2\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 parabla. This can be done by substituting values of $x$ and solving the corresponding value of $y$. Let $y=f(x).$ Then $
y=-2x^2+8x-5
$. Substituting values of $x$ and solving $y$ results to
\begin{array}{l|r}
\text{If }x=0: & \text{If }x=1:
\\\\
y=-2(0)^2+8(0)-5 & y=-2(1)^2+8(1)-5
\\
y=-2(0)+8(0)-5 & y=-2(1)+8(1)-5
\\
y=0+0-5 & y=-2+8-5
\\
y=-5 & y=1
.\end{array}
Hence, the points $
(0,-5)
$ and $
(1,1)
$ are on the parabola. Reflecting these points about the axis of symmetry, the points $
(3,1)
$ and $
(4,-5)
$ are also on the parabola.
Using the points $\{
(0,-5), (1,1),
\left(2,3\right),
(3,1), (4,-5)
\}$ 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\}
$.
