document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=206971>Question 206971</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If k(x)=f(-x), where would coordinates shift on a graph? Up or down, left or right? thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #156496 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=156496\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If we're given the graph of f(x) and then we replace EVERY x value with its opposite value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then what simply happens is that the graph will reflect over the y-axis. Why? Think of a number line. Draw it out if you have to. Now let's say that we plotted a bunch of random points on this number line. These points are sort of like the graph. Now negate each point. So a point of x=2 will become x=-2 and something like x=-4 will then become x=4. What you will see happening is that the points will simply reflect over the middle x=0. \r<BR>\n" );
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document.write( "If you add another axis (the y-axis), this works the same way. However, this time the points in question are connected by a continuous line or curve. \r<BR>\n" );
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document.write( "To answer your question, the graph doesn't shift since you aren't doing any translations. The graph is reflecting over the y-axis.\r<BR>\n" );
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document.write( "Here are some examples:\r<BR>\n" );
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document.write( "Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E3 ALIGN=MIDDLE  ALT=\"y=x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2Cx%5E3%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "compare that to the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-x%29%5E3 ALIGN=MIDDLE  ALT=\"y=%28-x%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Note: Just replace every 'x' with '-x'\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%28-x%29%5E3%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we graph the two together, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2Cx%5E3%2C%28-x%29%5E3%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E3 ALIGN=MIDDLE  ALT=\"y=x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-x%29%5E3 ALIGN=MIDDLE  ALT=\"y=%28-x%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green)\r<BR>\n" );
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document.write( "From the last graph, we see that the second graph is simply a reflection of the first one.<BR>\n" );
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