Question 1055984
(a)
The reflection about {{{ y = 4 }}} ( a horizontal line )
Must be {{{ 4 + 4 = 8 }}} units along y-axis from ( 0,0 )
which {{{ y = 2x }}} goes through
This point is ( 0, 8 )
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The intercept at {{{ y = 4 }}} is 
{{{ y = 2x }}}
{{{ 4 = 2x }}}
{{{ x = 2 }}}
( 2,4 ) , which stays the same for reflected line
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The slope goes through these 2 points and must have
the negative of the slope of the given line.
Now use the general point-slope formula
{{{ ( y - 4 ) / ( x - 2 ) = ( 8 - 4 ) / ( 0 - 2 ) }}}
{{{ ( y - 4 ) / ( x - 2 ) = 4 / (-2) }}}
{{{ ( y - 4 ) / ( x - 2 ) = -2 }}}
{{{ y - 4 = -2*( x - 2 ) }}}
{{{ y - 4 = -2x + 4 }}}
{{{ y = -2x + 8 }}}
( note that I get the correct slope, {{{ -2 }}} )
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Here's the plot of the 2 lines:
{{{ graph( 400, 400, -10, 10, -10, 10, 2x, 4, -2x + 8 ) }}}
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(b)
Reflected about {{{ x = 2 }}}
This is a vertical line through ( 2, 0 )
The reflection of ( 0,0 ) would be ( 4,0 )
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{{{ y = 2x }}} intercepts {{{ x=2 }}} at ( 2,4 )
The reflection is at the same point ( 2,4 )
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Again, I should end up with {{{ -2 }}} for a slope
Use general point-slope formula
{{{ ( y - 0 ) / ( x - 4 ) = ( 4 - 0 ) / ( 2 - 4 ) }}}
{{{ y / ( x - 4 ) = 4/(-2) }}}
{{{ y / ( x - 4 ) = -2 }}}
{{{ y = -2*( x - 4 ) }}}
{{{ y = -2x + 8 }}}
This is the same line as in previous problem.
so, it doesn't matter whether you use {{{ y = 4 }}}
of {{{ x = 2 }}} for a reflection.
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Definitely get a 2nd opinion on this, it looks right
to me, but it's a little tricky