document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199243>Question 1199243</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The graph of a basic cubic function k(x)=x^3 is shown.  Suppose that p(x)=k(x+3).  Use reference points and symmetry to complete the table of values for p(x).  Then graph p(x) on the same coordinate plane as k(x) and label it.    \r<BR>\n" );
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document.write( "\Reference points k(x) given<BR>\n" );
document.write( "(0,0)<BR>\n" );
document.write( "(1,1)<BR>\n" );
document.write( "(2,8)\r<BR>\n" );
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document.write( "I am having a hard time with the substitution -I can do the graph once I have the substitution done<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833017 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=833017\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Here's the graph of y = x^3 through the three reference points mentioned.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-3%2C10%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-3%2C10%2C-100%2Cx%5E3%29%2C%0D%0Acircle%280%2C0%2C0.05%29%2C%0D%0Acircle%280%2C0%2C0.07%29%2C%0D%0Acircle%280%2C0%2C0.08%29%2C%0D%0A%0D%0Acircle%281%2C1%2C0.05%29%2C%0D%0Acircle%281%2C1%2C0.07%29%2C%0D%0Acircle%281%2C1%2C0.08%29%2C%0D%0A%0D%0Acircle%282%2C8%2C0.05%29%2C%0D%0Acircle%282%2C8%2C0.07%29%2C%0D%0Acircle%282%2C8%2C0.08%29%2C%0D%0A%0D%0A%0D%0Alocate%280%2C0%2C%22%280%2C0%29%22%29%2C%0D%0Alocate%281%2C1%2C%22%281%2C1%29%22%29%2C%0D%0Alocate%282%2C8%2C%22%282%2C8%29%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Imagine this green curve is etched in stone on a wall or the ground.<BR>\n" );
document.write( "Being etched in stone means the green curve itself physically cannot be moved.\r<BR>\n" );
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document.write( "Imagine that what can move however is the red xy axis. <BR>\n" );
document.write( "Think of this axis as part of the camera looking at the curve. <BR>\n" );
document.write( "We can think of this as the crosshairs of the camera.\r<BR>\n" );
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document.write( "Since you're holding the camera, you can of course move the camera left, right, up, or down.\r<BR>\n" );
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document.write( "If you moved the camera's crosshairs 3 units to the right, then each x input is now 3 units larger.<BR>\n" );
document.write( "In other words: the old input x is now x+3\r<BR>\n" );
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document.write( "Example: if x = 1 was the old input, then x+3 = 1+3 = 4 is the new input.\r<BR>\n" );
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document.write( "After the crosshairs move 3 units to the right, it gives the illusion the green curve (remember that it's etched in stone) has moved 3 units to the left. <BR>\n" );
document.write( "It's all a matter of perspective more or less.\r<BR>\n" );
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document.write( "Effectively, the jump from k(x) = x^3 to p(x) = k(x+3) will move this green curve 3 units to the left. <BR>\n" );
document.write( "We'll arrive at the blue curve as shown below.\r<BR>\n" );
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document.write( "All points on this curve also move 3 units to the left. After all, a curve is simply a collection of points.\r<BR>\n" );
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document.write( "The point (0,0) moves to (-3,0)<BR>\n" );
document.write( "The point (1,1) moves to (-2,1)<BR>\n" );
document.write( "The point (2,8) moves to (-1,8)<BR>\n" );
document.write( "We subtract 3 from the x coordinate. <BR>\n" );
document.write( "Keep the y coordinate the same.\r<BR>\n" );
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document.write( "Here's what this shifting looks like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-3%2C10%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-3%2C10%2C-100%2Cx%5E3%2C%28x%2B3%29%5E3%29%2C%0D%0A%0D%0Acircle%280%2C0%2C0.05%29%2C%0D%0Acircle%280%2C0%2C0.07%29%2C%0D%0Acircle%280%2C0%2C0.08%29%2C%0D%0Acircle%281%2C1%2C0.05%29%2C%0D%0Acircle%281%2C1%2C0.07%29%2C%0D%0Acircle%281%2C1%2C0.08%29%2C%0D%0Acircle%282%2C8%2C0.05%29%2C%0D%0Acircle%282%2C8%2C0.07%29%2C%0D%0Acircle%282%2C8%2C0.08%29%2C%0D%0Alocate%280%2C0%2C%22%280%2C0%29%22%29%2C%0D%0Alocate%281%2C1%2C%22%281%2C1%29%22%29%2C%0D%0Alocate%282%2C8%2C%22%282%2C8%29%22%29%2C%0D%0A%0D%0A%0D%0Acircle%28-3%2C0%2C0.05%29%2C%0D%0Acircle%28-3%2C0%2C0.07%29%2C%0D%0Acircle%28-3%2C0%2C0.08%29%2C%0D%0Acircle%28-2%2C1%2C0.05%29%2C%0D%0Acircle%28-2%2C1%2C0.07%29%2C%0D%0Acircle%28-2%2C1%2C0.08%29%2C%0D%0Acircle%28-1%2C8%2C0.05%29%2C%0D%0Acircle%28-1%2C8%2C0.07%29%2C%0D%0Acircle%28-1%2C8%2C0.08%29%2C%0D%0Alocate%28-4%2C0.8%2C%22%28-3%2C0%29%22%29%2C%0D%0Alocate%28-2%2C1%2C%22%28-2%2C1%29%22%29%2C%0D%0Alocate%28-2.5%2C8%2C%22%28-1%2C8%29%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "k(x) = x^3 in green<BR>\n" );
document.write( "p(x) = k(x+3) = (x+3)^3 in blue\r<BR>\n" );
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document.write( "You can use free graphing software tools like Desmos or GeoGebra to confirm the above graphs.\r<BR>\n" );
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document.write( "In short:\r<BR>\n" );
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document.write( "The jump from k(x) to k(x+3) means we shift the entire curve 3 units to the left.\r<BR>\n" );
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document.write( "A point like (0,0) moves to (-3,0) after such a shift. Use this idea for the other reference points as well.\r<BR>\n" );
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document.write( "Let me know if you have any questions.<BR>\n" );
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