Answer
The $y-$ intercept is $1$.
The equation $y={{x}^{4}}+2{{x}^{2}}+1$ is symmetric with respect to the $y-$axis.
Work Step by Step
To find the $x-$ intercept(s),
First take $y=0$ in the equation, then solve for $x$, where $x$ is a real number.
Here, $y={{x}^{4}}+2{{x}^{2}}+1$.
If $y=0$, then ${{x}^{4}}+2{{x}^{2}}+1=0$.
By simplifying for $x$,
Substitute ${{x}^{2}}=u$ $\Rightarrow {{x}^{4}}={{u}^{2}}$.
${{u}^{2}}+2u+1=0$.
By factoring,
$\left( u+1 \right)\left( u+1 \right)=0$.
By zero-product property,
$\Rightarrow \left( u+1 \right)=0$,
$\Rightarrow u=-1$.
By resubstituting the value $u=-1$ in the equation ${{x}^{2}}=u$,
$\Rightarrow {{x}^{2}}=-1$.
Square root both sides,
$\Rightarrow x=\pm \sqrt{-1}$,
$\Rightarrow x=\pm i$,
$\Rightarrow x=i\,$or $x=-i\,$.
The equation has two solutions, $x=i\,$ and $x=-i\,$, but these solutions are not real numbers.
Therefore, the equation $y={{x}^{4}}+2{{x}^{2}}+1$ has no $x-$ intercept.
To find the $y-$ intercept,
First take $x=0$ in the equation, then solve for $y$, where $y$ is a real number.
Let $x=0$ in the equation,
$y={{\left( 0 \right)}^{4}}+2{{\left( 0 \right)}^{2}}+1$
$=1$
$\Rightarrow y=1$
The $y-$ intercept is $1$.
To test for symmetry with respect to the $x-$axis, replace $y$ by $-y$ in the equation $y={{x}^{4}}+2{{x}^{2}}+1$,
$\Rightarrow \left( -y \right)={{x}^{4}}+2{{x}^{2}}+1$,
$\Rightarrow y=-{{x}^{4}}-2{{x}^{2}}-1$.
The equation $y=-{{x}^{4}}-2{{x}^{2}}-1$ is not equivalent to $y={{x}^{4}}+2{{x}^{2}}+1$.
Therefore, the graph of the equation is not symmetric with respect to the $x-$axis.
To test for symmetry with respect to the $y-$axis, replace $x$ by $-x$ in the equation $y={{x}^{4}}+2{{x}^{2}}+1$,
$\Rightarrow y={{\left( -x \right)}^{4}}+2{{\left( -x \right)}^{2}}+1$,
$\Rightarrow y={{\left( x \right)}^{4}}+2{{\left( x \right)}^{2}}+1$.
The equation $y={{\left( -x \right)}^{4}}+2{{\left( -x \right)}^{2}}+1$ is equivalent to $y={{x}^{4}}+2{{x}^{2}}+1$.
Therefore, the graph of the equation is symmetric with respect to the $y-$axis.
To test for symmetry with respect to the origin, replace $x$ by $-x$ and replace $y$ by $-y$ in the equation $y={{x}^{4}}+2{{x}^{2}}+1$,
$\Rightarrow \left( -y \right)={{\left( -x \right)}^{4}}+2{{\left( -x \right)}^{2}}+1$
$\Rightarrow -y={{\left( x \right)}^{4}}+2{{\left( x \right)}^{2}}+1$
The equation $-y={{\left( x \right)}^{4}}+2{{\left( x \right)}^{2}}+1$ is not equivalent to $y={{x}^{4}}+2{{x}^{2}}+1$.
Therefore, the equation $y={{x}^{4}}+2{{x}^{2}}+1$ is symmetric only with respect to the $y-$axis.
