Question 659376
{{{y=x-3}}}


1. symmetry about the {{{y-axis}}}

For symmetry about the {{{y-axis}}}, check to see if it is the same when you replace {{{x}}} with {{{-x}}}

{{{-x-3 = x-3}}}
{{{-1-3 = 1-3}}}
{{{-4 <> -2}}}
 
So {{{y = x-3}}} does {{{NOT}}} have symmetry about the {{{y-axis}}}

2. symmetry about the {{{x-axis}}}

For symmetry about the {{{x-axis}}}, check to see if it is the same when you replace {{{y}}} with {{{-y}}}

{{{-y = -x+3}}} &#8800; {{{x-3}}}

So {{{y = x-3}}} does {{{NOT}}} have symmetry about the {{{x-axis}}}

3. diagonal symmetry

For diagonal symmetry, check to see if it is the same when you swap {{{y}}} and {{{x}}}:

{{{x=y-3}}} ...=>.. is not the same as {{{y=x-3}}}

So {{{y=x-3}}} does {{{NOT}}} have diagonal symmetry


4. origin symmetry

Start with:
{{{y=x-3}}} 


Replace {{{x}}} with {{{-x}}} and {{{y}}} with {{{-y}}}:

{{{-y=-x-3}}} 

Multiply both sides by {{{-1}}}:

{{{-y*-1=-x*(-1)-3 (-1)}}}

{{{y=x+3 }}} 

we have the equation {{{y=x+3}}} which is {{{not}}} equal to original equation {{{y=x-3}}} , hence {{{ y=x-3}}}  does {{{NOT}}} have origin symmetry


{{{ graph( 600, 600, -6, 5, -10, 10, x-3) }}}