Question 278070
Write an equation for a function that is the graph of  {{{y=1/x^3}}} reflected through the y-axis and shifted down 3 units,

We start with the graph of {{{y=1/x^3}}}

{{{graph(400,400,-10,10,-10,10,11,1/x^3)}}}

It has the x and the y-axis as asymptotes

Then we reflect it through the y-axis by replacing {{{x}}} by {{{-x}}}.

{{{y=1/x^3}}} becomes

{{{y=1/(-x)^3}}}

which simplifies to:

{{{y=-1/x^3}}}

This gives the graph below

{{{graph(400,400,-10,10,-10,10,11,-1/x^3)}}}

with the same two asymptotes.

Finally we shift it down 3 units by subtracting 3 from the
right side, which gives the equation

{{{y=-1/x^3-3}}}}

and the graph is:

{{{graph(400,400,-10,10,-10,10,11,-1/x^3-3,-3sqrt(sin(5x))/sqrt(sin(5x))))}}}

The blue dotted line is the asymptote.  The original curve had
the x-axis and y-axis as its asymptotes, and this new graph has the line y=-3 as its horizontal asymptote, and has the y-axis as its vertical asymptote..

Edwin</pre>