Question 214873
{{{ y = abs(2x - 6) - 2}}}
<pre><font size = 4 color = "indigo"><b>
To find the vertex:
Set the expression between the absolute value bars
equal to zero, that's just the red part below:

{{{ y = abs(red(2x - 6)) - 2}}}

{{{red(2x-6)=0}}}
{{{2x=6}}}
{{{x=3}}}

Now find the y-value that corresponds to that
value of x:

{{{ y = abs(2(3) - 6) - 2}}}
{{{ y = abs(6 - 6) - 2}}}
{{{ y = abs(0) - 2}}}
{{{ y = 0 - 2}}}
{{{ y = -2}}}

So the vertex is the point {{{"(3,-2)"}}}}

So we plot that point:

{{{drawing(200,200,-5,7,-6,6, 
graph(200,200,-5,7,-6,6)
line(3+.1,-2,3-.1,-2), line(3,-2+.1,3,-2-.1), line(3+.1,-2+.1,3-.1,-2-.1), line(3+.1,-2-.1,3-.1,-2+.1)  )}}} 

Now find and plot one point on each side of that point:

So choose x to be 1 less than 3, which is 2, Substitute
in:

{{{ y = abs(2x - 6) - 2}}}
{{{ y = abs(2(2) - 6) - 2}}}
{{{ y = abs(4 - 6) - 2}}}
{{{ y = abs(-2) - 2}}}
{{{ y = 2 - 2}}}
{{{ y = 0}}}

So we plot the point {{{"(2,0)"}}}

{{{drawing(200,200,-5,7,-6,6, 
graph(200,200,-5,7,-6,6),
line(2+.1,0,2-.1,0), line(2,0+.1,2,0-.1), line(2+.1,0+.1,2-.1,0-.1), line(2+.1,0-.1,2-.1,0+.1),

line(3+.1,-2,3-.1,-2), line(3,-2+.1,3,-2-.1), line(3+.1,-2+.1,3-.1,-2-.1), line(3+.1,-2-.1,3-.1,-2+.1)  )}}} 

Next choose x to be 1 more than 3, which is 4, Substitute
in:

{{{ y = abs(2x - 6) - 2}}}
{{{ y = abs(2(4) - 6) - 2}}}
{{{ y = abs(8 - 6) - 2}}}
{{{ y = abs(2) - 2}}}
{{{ y = 2 - 2}}}
{{{ y = 0}}}

So we plot the point {{{"(4,0)"}}}

{{{drawing(200,200,-5,7,-6,6, 
graph(200,200,-5,7,-6,6),
line(2+.1,0,2-.1,0), line(2,0+.1,2,0-.1), line(2+.1,0+.1,2-.1,0-.1), line(2+.1,0-.1,2-.1,0+.1),

line(4+.1,0,4-.1,0), line(4,0+.1,4,0-.1), line(4+.1,0+.1,4-.1,0-.1), line(4+.1,0-.1,4-.1,0+.1),


line(3+.1,-2,3-.1,-2), line(3,-2+.1,3,-2-.1), line(3+.1,-2+.1,3-.1,-2-.1), line(3+.1,-2-.1,3-.1,-2+.1)  )}}}

Then we draw the v-shaped graph:

{{{drawing(200,200,-5,7,-6,6, 
graph(200,200,-5,7,-6,6,abs(2x - 6) - 2),
line(2+.1,0,2-.1,0), line(2,0+.1,2,0-.1), line(2+.1,0+.1,2-.1,0-.1), line(2+.1,0-.1,2-.1,0+.1),  

line(4+.1,0,4-.1,0), line(4,0+.1,4,0-.1), line(4+.1,0+.1,4-.1,0-.1), line(4+.1,0-.1,4-.1,0+.1),


line(3+.1,-2,3-.1,-2), line(3,-2+.1,3,-2-.1), line(3+.1,-2+.1,3-.1,-2-.1), line(3+.1,-2-.1,3-.1,-2+.1)  )}}}

That's it.

Edwin</pre>