document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1080930>Question 1080930</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sketch the curve represented by the parametric equations. Then eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Thanks so much.\r<BR>\n" );
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document.write( "Sketch and Equations: https://s21.postimg.org/5tmtsylon/x_and_y_and_sketch.png<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 #695078 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=695078\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> that straight line is extraneous and doesn't belong there.\r<BR>\n" );
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document.write( "in fact, that straight line is based on the equation of y = 1/2 * x -1\r<BR>\n" );
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document.write( "i did show it on the graph to confirm that the line was, in fact, based on that equation.\r<BR>\n" );
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document.write( "then i removed it.\r<BR>\n" );
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document.write( "your parametric equation is:\r<BR>\n" );
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document.write( "x = t^2 - 6<BR>\n" );
document.write( "y = 1/2 * t - 1\r<BR>\n" );
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document.write( "in the desmos.com calculator, this is graphed as (t^2-6,1/2*t-1)\r<BR>\n" );
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document.write( "that calculator can be found here:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator\" target = \"_blank\">https://www.desmos.com/calculator</a>\r<BR>\n" );
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document.write( "that means the x-coordinate on the graph is based on x = t^2 - 6 and the y-coordinate on the graph is based on y = 1/2 * t - 1\r<BR>\n" );
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document.write( "you do not see the value of t on the graph.\r<BR>\n" );
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document.write( "you only see the coordinate values of (x,y).\r<BR>\n" );
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document.write( "to convert from the parametric equation of x = t^2 - 6 and y = 1/2 * t - 1, you can do the f0llowing:\r<BR>\n" );
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document.write( "start with x = t^2 - 6<BR>\n" );
document.write( "add 6 to both sides to get x + 6 = t^2<BR>\n" );
document.write( "solve for t to get t = plus or minus sqrt(x+6)\r<BR>\n" );
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document.write( "in the equation of y = 1/2 * t - 1, replace t with plus or minus sqrt(x+6) to get:\r<BR>\n" );
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document.write( "when plus, y = 1/2 * t - 1 becomes y = 1/2 * sqrt(x+6) - 1\r<BR>\n" );
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document.write( "when minus, y = 1/2 * t - 1 becomes y = 1/2 * -sqrt(x+6) - 1 which becomes y = -1/2 * sqrt(x+6) - 1\r<BR>\n" );
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document.write( "your converted equation is therefore in two parts.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "y = 1/2 * sqrt(x + 6) - 1 and y = -1/2 * sqrt(x + 6) - 1\r<BR>\n" );
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document.write( "the parametric equation and these translated equations are all shown in blue on the graph.\r<BR>\n" );
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document.write( "all 3 of them generate the same graph because the parametric equations is equivalent to the two y = ... equations.\r<BR>\n" );
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document.write( "when generating the parametric equations in the desmos.com calculator, you need to supply the limits for the value of t.\r<BR>\n" );
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document.write( "i made t go from -10 to 10 which was enough to show the graph in a similar manner as depicted on the image you provided.\r<BR>\n" );
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document.write( "the graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/051305.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "i also created a table of values based on the parametric equation that included the value of x and y when t = 0 and when t = -6 and when t = 6.\r<BR>\n" );
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document.write( "that table of values is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/051306.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "you can see from the table, that when t = 0, x = -6 and y = -1.\r<BR>\n" );
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document.write( "you can also see from the table, that when t = -6, x = 30 and y = -4.\r<BR>\n" );
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document.write( "you can also see from the table, that when t = 6, x = 30 and y = 2.\r<BR>\n" );
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document.write( "all these values can also be seen on the graph.\r<BR>\n" );
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document.write( "you see (-6,-1), (30,-4) and (30,2)\r<BR>\n" );
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document.write( "(-6,-1) is when t = 0<BR>\n" );
document.write( "(30,-4) is when t = -6<BR>\n" );
document.write( "(30,2)is when t = 6\r<BR>\n" );
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