Answer
Graph the function as:
Work Step by Step
Step 1: Substitute $-x$ in place of x.
$\begin{align}
& f\left( x \right)=\frac{{{x}^{2}}-4x+3}{{{\left( x+1 \right)}^{2}}} \\
& f\left( -x \right)=\frac{{{\left( -x \right)}^{2}}-4\left( -x \right)+3}{{{\left( \left( -x \right)+1 \right)}^{2}}} \\
& =\frac{{{x}^{2}}+4x+3}{{{\left( 1-x \right)}^{2}}} \\
& \ne -f\left( x \right)
\end{align}$
And,
$\begin{align}
& -f\left( x \right)=-\left( \frac{{{x}^{2}}-4x+3}{{{\left( x+1 \right)}^{2}}} \right) \\
& \ne f\left( -x \right)
\end{align}$
Hence the graph of the function is symmetric neither about the $y$-axis nor origin.
Step 2: To calculate the x intercepts eqaute $f\left( x \right)=0$.
$\begin{align}
& \frac{{{x}^{2}}-4x+3}{{{\left( x+1 \right)}^{2}}}=0 \\
& x=1,3
\end{align}$ ,
Step 3: To calculate the y intercept evaluate $f\left( 0 \right)$
$f\left( 0 \right)=3$
Step 4: Since the degree of the numerator is equal to the denominator, the horizontal asymptote is:
$y=1$
Step 5: For the vertical asymptote, equate the denominator to 0.
$x=-1$
