Answer

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**Hint:**Type of question is based on the concept of finding symmetry algebraically, which says that If we want to find any graph symmetric to x-axis, then replace ‘y’ with ‘-y’ and on simplifying if we get same equation we have in initial then it is symmetric, otherwise not. Same it with for symmetric about y-axis, replace ‘x’ with ‘-x’ if we get the equation we have initially then it is symmetric otherwise not. And to check for symmetry about origin replace both ‘x’ and ‘y’ with ‘-x’ and ‘-y’ respectively and on simplifying we get the equation we have initially then it is symmetric about origin otherwise not.

**Complete step-by-step solution:**

Moving ahead with the question, first let us check symmetricity for $y=|x|+5$;

Check for symmetric about y-axis;

Replace ‘x’ with ‘-x’ then we will get;

$\begin{align}

& y=|x|+5 \\

& y=|-x|+5 \\

\end{align}$

As we know that $|-a|=|a|$, in which ‘||’ represents mode.

So we will get;

$y=|x|+5$

Hence the graph is symmetric about the y-axis.

Check for symmetric about x-axis;

Replace ‘y’ with ‘-y’ then we will get;

$\begin{align}

& y=|x|+5 \\

& -y=|x|+5 \\

\end{align}$

So we will get;

$-y=|x|+5$

As this equation is not same as $y=|x|+5$, so we can say that the graph is not symmetric about the x-axis.

Similarly check for symmetric about origin;

Replace both ‘x’ and ‘y’ with ‘-x’ and ‘-y’ respectively

So we will get;

$\begin{align}

& y=|x|+5 \\

& -y=|-x|+5 \\

\end{align}$

Which we can reduce it to;

$-y=|x|+5$

As this equation is not same as $y=|x|+5$, so we can say that the graph is not symmetric about origin.

B). Now let us check symmetricity about $xy=4$;

Check for symmetric about y-axis;

Replace ‘x’ with ‘-x’ then we will get;

$\begin{align}

& xy=4 \\

& \left( -x \right)y=4 \\

\end{align}$

So we can reduce it to;

$-xy=4$

As this equation is not the same as $xy=4$, hence the graph is not symmetric about y-axis.

Check for symmetric about x-axis;

Replace ‘y’ with ‘-y’ then we will get;

$\begin{align}

& xy=4 \\

& x\left( -y \right)=4 \\

\end{align}$

So we can reduce it to;

$-xy=4$

As this equation is not the same as $xy=4$, so we can say that the graph is not symmetric about the x-axis.

Similarly check for symmetric about origin;

Replace both ‘x’ and ‘y’ with ‘-x’ and ‘-y’ respectively

So we will get;

$\begin{align}

& xy=4 \\

& \left( -x \right)\left( -y \right)=4 \\

\end{align}$

SO we can reduce it to;

$xy=4$

As this equation is the same as $xy=4$, so we can say that the graph is symmetric about origin.

C). Now let us check symmetricity for $x+{{y}^{2}}=8$;

Check for symmetric about y-axis;

Replace ‘x’ with ‘-x’ then we will get;

$\begin{align}

& x+{{y}^{2}}=8 \\

& \left( -x \right)+{{y}^{2}}=8 \\

\end{align}$

So we can reduce it to;

$-x+{{y}^{2}}=8$

As the equation is not the same as $x+{{y}^{2}}=8$ which is initially one, Hence the graph is not symmetric about the y-axis.

Check for symmetric about x-axis;

Replace ‘y’ with ‘-y’ then we will get;

$\begin{align}

& x+{{y}^{2}}=8 \\

& x+{{\left( -y \right)}^{2}}=8 \\

\end{align}$

So we will get;

$x+{{y}^{2}}=8$

As this equation is the same as $x+{{y}^{2}}=8$, so we can say that the graph is symmetric about the x-axis.

Similarly check for symmetric about origin;

Replace both ‘x’ and ‘y’ with ‘-x’ and ‘-y’ respectively

So we will get;

$\begin{align}

& x+{{y}^{2}}=8 \\

& \left( -x \right)+{{\left( -y \right)}^{2}}=8 \\

\end{align}$

Which we can reduce it to;

$-x+{{y}^{2}}=8$

As this equation is not the same as $x+{{y}^{2}}=8$, so we can say that the graph is not symmetric about origin.

Hence from above result we can say that the equation $y=|x|+5$ is symmetric about x-axis, the equation $xy=4$ is symmetric about origin and the equation $x+{{y}^{2}}=8$ is symmetric about x-axis.

**Note:**This is the algebraic process of finding out the equation whether it is symmetric about origin, x-axis or y-axis. Otherwise we can find the symmetry by drawing the equation on the cartesian plane.

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