document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164851>Question 1164851</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What must be done to a function's equation so that its graph is shifted vertically upward? Compare this to what must be done so that its graph is shifted horizontally to the right. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #789295 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=789295\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> to shift the graph vertically upward, you need add a constant to the end of it.\r<BR>\n" );
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document.write( "so, if your equation is f(x), then your new equation has to be f(x) + a, where is a positive constant.\r<BR>\n" );
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document.write( "to shift the graph horizontally to the right, you need to subtract a constant from the variable used.\r<BR>\n" );
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document.write( "so, if your equation is f(x), then your new equation has to be f(x-a), where a is a positive constant.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "consider the equation of f(x) = x^2\r<BR>\n" );
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document.write( "f(x) + 5 = x^2 + 5 shifts the equation up 5 units.\r<BR>\n" );
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document.write( "this means that the value of y for the same value of x is 5 units more positive than the original equation.\r<BR>\n" );
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document.write( "you can see on the graph, that, in the equation of y = x^2, y = 0 when x = 0.\r<BR>\n" );
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document.write( "you can also see on the graph, that, in the equation of y = x^2 + 5, y = 5 when x = 0.\r<BR>\n" );
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document.write( "in the graph, y = x^2 is red and y = x^2 + 5 is blue.\r<BR>\n" );
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document.write( "now consider the equation of  f(x) = x^2 again.\r<BR>\n" );
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document.write( "f(x-5) = (x-5)^2 shift the equation to the right  5 units.\r<BR>\n" );
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document.write( "this means that the value of x for the same value of y is 5 units to the right of the original equation.\r<BR>\n" );
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document.write( "you can see on the graph that, in the equation of y = x^2, x = 0 when y = 0.\r<BR>\n" );
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document.write( "you can also see on the graph that, in the equation of y = (x-5)^2, x = 5 when y = 0.\r<BR>\n" );
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document.write( "in the graph, y = x^2 is red and y = (x-5)^2 is green.\r<BR>\n" );
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document.write( "here's the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://theo.x10hosting.com/2020/091401.jpg\" >\r<BR>\n" );
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document.write( "note that the use of f(x) and the use of y is interchangeable.\r<BR>\n" );
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document.write( "that's because f(x) = y.\r<BR>\n" );
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document.write( "you can say y = x^2 and you can say f(x) = x^2 and it means the same thing.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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