document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151870>Question 1151870</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the equation formed from the families of straight lines tangent to the parabola <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2=2x ALIGN=MIDDLE  ALT=\"y%5E2=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773762 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=773762\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2=2x ALIGN=MIDDLE  ALT=\"y%5E2=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is symmetrical with respect to the x-axis.  So we can find the equations of the tangent lines for the portion of the graph with y positive; then any tangent line with equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > will have a corresponding tangent line with equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%28mx%2Bb%29 ALIGN=MIDDLE  ALT=\"y=-%28mx%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-mx-b ALIGN=MIDDLE  ALT=\"y=-mx-b\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The equation for the portion of the graph with y positive is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%282x%29+=+%282x%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%282x%29+=+%282x%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Take the derivative, using the power and chain rules, to find the slope:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=%281%2F2%29%282x%29%5E%28-1%2F2%29%282%29+=+%282x%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=%281%2F2%29%282x%29%5E%28-1%2F2%29%282%29+=+%282x%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For each value of x, the tangent line will have slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"%282x%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; and the y value will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"%282x%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For a given value of x, find the expression for the y-intercept b for the tangent line with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%282x%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"y+=+%282x%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%29%5E%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"%282x%29%5E%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+mx%2Bb ALIGN=MIDDLE  ALT=\"y+=+mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%29%5E%281%2F2%29+=+%28%282x%29%5E%28-1%2F2%29%29x+%2B+b ALIGN=MIDDLE  ALT=\"%282x%29%5E%281%2F2%29+=+%28%282x%29%5E%28-1%2F2%29%29x+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2Asqrt%28x%29+=+%281%2F%28sqrt%282%29%2Asqrt%28x%29%29%29%2Ax+%2B+b ALIGN=MIDDLE  ALT=\"sqrt%282%29%2Asqrt%28x%29+=+%281%2F%28sqrt%282%29%2Asqrt%28x%29%29%29%2Ax+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+sqrt%282%29%2Asqrt%28x%29+-+%281%2F%28sqrt%282%29%2Asqrt%28x%29%29%29%2Ax ALIGN=MIDDLE  ALT=\"b+=+sqrt%282%29%2Asqrt%28x%29+-+%281%2F%28sqrt%282%29%2Asqrt%28x%29%29%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+sqrt%282%29%2Asqrt%28x%29+-+%28sqrt%28x%29%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"b+=+sqrt%282%29%2Asqrt%28x%29+-+%28sqrt%28x%29%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+sqrt%28x%29%28sqrt%282%29-1%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"b+=+sqrt%28x%29%28sqrt%282%29-1%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+sqrt%28x%29%282%2Fsqrt%282%29-1%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"b+=+sqrt%28x%29%282%2Fsqrt%282%29-1%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+sqrt%28x%29%281%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"b+=+sqrt%28x%29%281%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+sqrt%28x%2F2%29 ALIGN=MIDDLE  ALT=\"b+=+sqrt%28x%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So for a given value of x, the tangent line in the first quadrant has slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%282x%29 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and y-intercept <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The equation is then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%281%2Fsqrt%282x%29%29x+%2B+sqrt%28x%2F2%29 ALIGN=MIDDLE  ALT=\"y+=+%281%2Fsqrt%282x%29%29x+%2B+sqrt%28x%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Choosing some \"nice\" values of x that make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x%29 ALIGN=MIDDLE  ALT=\"sqrt%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > rational....<br><BR>\n" );
document.write( "x = 1/2:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%2B1%2F2 ALIGN=MIDDLE  ALT=\"y+=+x%2B1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 2:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%281%2F2%29x%2B1 ALIGN=MIDDLE  ALT=\"y+=+%281%2F2%29x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 9/2:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%281%2F3%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y+=+%281%2F3%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 8:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%281%2F4%29%2B2 ALIGN=MIDDLE  ALT=\"y+=+%281%2F4%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And then there are corresponding tangent lines on the other branch of the parabola:<br><BR>\n" );
document.write( "x = 1/2:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-x-1%2F2 ALIGN=MIDDLE  ALT=\"y+=+-x-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 2:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1%2F2%29x-1 ALIGN=MIDDLE  ALT=\"y+=+%28-1%2F2%29x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 9/2:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1%2F3%29x-3%2F2 ALIGN=MIDDLE  ALT=\"y+=+%28-1%2F3%29x-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 8:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1%2F4%29-2 ALIGN=MIDDLE  ALT=\"y+=+%28-1%2F4%29-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph of the parabola and the tangent lines for x=1/2 -- x+1/2 and -x-1/2:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C400%2C-2%2C10%2C-5%2C5%2Csqrt%282x%29%2C-sqrt%282x%29%2Cx%2B1%2F2%2C-x-1%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A graph of the parabola and the tangent lines for x=2 -- (1/2)x+1 and (-1/2)x-1:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C400%2C-2%2C10%2C-5%2C5%2Csqrt%282x%29%2C-sqrt%282x%29%2C%281%2F2%29x%2B1%2C%28-1%2F2%29x-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A graph of the parabola and the tangent lines for x=8 -- (1/4)x+2 and (-1/4)x-2:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C400%2C-2%2C10%2C-5%2C5%2Csqrt%282x%29%2C-sqrt%282x%29%2C%281%2F4%29x%2B2%2C%28-1%2F4%29x-2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );