<PRE><B>Question 979372</B>
 
a. {{{y=2x^4-3}}}

We can often see symmetry visually, but to be really sure we should check a simple fact:

For symmetry with respect to the y-axis, check to see if the equation is the same when we replace {{{x}}} with {{{-x}}}:


 is  {{{y=2x^4-3}}} symmetric about the y-axis?

Try to replace {{{x}}} with {{{-x}}}:

{{{y=2(-x)^4-3}}}

Since {{{(-x)^4 = x^4}}} (multiplying a negative times a negative gives a positive), there is {{{no}}} change.

So, {{{y=2x^4-3 }}} {{{is}}} symmetric about the y-axis


For symmetry about x-axis use the same idea as for the y-axis, but try replacing {{{y}}} with {{{-y}}}.

{{{-y=2x^4-3 }}}
{{{-y/-1=2x^4/-1-3/-1 }}}

{{{y=-2x^4+3 }}}

Now try to get the original equation:

{{{2x^4-3&lt;&gt;-2x^4+3 }}} =&gt; It is {{{different}}}.

Hence {{{y=2x^4-3 }}} is {{{not}}} symmetric about the x-axis

 
Diagonal symmetry:

Try swapping {{{y}}} and {{{x}}} :

{{{x=2y^4-3 }}} 

rearrange : 

{{{x+3=2y^4 }}}

{{{x/2+3/2=y^4 }}}

{{{y=root(4,x/2+3/2) }}}

And we do not have the original equation. They are not the same.

Hence {{{y=2x^4-3 }}} has {{{not}}} diagonal symmetry.


	
Origin symmetry is when every part has a matching part:

    the same distance from the central point
    but in the opposite direction.

Check to see if the equation is the same when we replace both {{{x}}} with {{{-x}}} and {{{y}}} with {{{-y}}}.

 does {{{y=2x^4-3 }}} have origin symmetry?

Start with:

{{{y=2x^4-3 }}}

Replace {{{x}}} with {{{-x}}} and {{{y}}} with {{{-y}}}:

{{{-y=2(-x)^4-3 }}}
{{{-y=2x^4-3 }}}

the left side is {{{different}}} from the original equation and the right side is identical to the original equation, and this isn’t equivalent to the original equation and we do {{{not}}} have symmetry about the origin 


so,  {{{y=2x^4-3 }}} is {{{not}}} symmetric about the origin


b. 
{{{y=x+15}}}
 to check is it symmetric about the y-axis, replace {{{x}}} with {{{-x}}}:

{{{y=-x+15}}}=&gt; is different from the original equation, so  it is not symmetric about the y-axis

to check is it symmetric about the x-axis, replace {{{y}}} with {{{-y}}}:

{{{-y=x+15}}}=&gt; is different from the original equation, so  it is not symmetric about the x-axis


To test for symmetry over the origin, you replace {{{x}}} with {{{-x}}}, and {{{y }}}with{{{ -y}}}.  
Again if the equation is identical to the original equation, then the graph is symmetric over the origin.
{{{-y=-x+15}}}=&gt;  the equation is not identical to the original equation, so  it is not symmetric over the origin

remember: The linear equations {{{y = x}}}  or {{{ y = -x}}}  are the only two equations which are symmetric over the origin.
Note that most graphs don’t have any kind of symmetry.  Also, it is possible for a graph to have more than one kind of symmetry.  For example the graph of a circle centered at the origin exhibits all three symmetries.

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