document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1035907>Question 1035907</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equations of the lines that pass through point (-1 , 3) and are tangent to the curve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=x%5E3+-+2x+%2B+1 ALIGN=MIDDLE  ALT=\"+y=x%5E3+-+2x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
document.write( "\n" );
document.write( "Graph the tangents and the curve </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #650529 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=650529\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let (a,b) be a point of tangency at the curve y, and let y - 3 = m(x+1) be the line of tangency passing through (-1,3), where m is the slope of the line.\r<BR>\n" );
document.write( "\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28b-3%29%2F%28a%2B1%29 ALIGN=MIDDLE  ALT=\"m+=+%28b-3%29%2F%28a%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slope of the tangent line.\r<BR>\n" );
document.write( "\n" );
document.write( "Now y' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+-+2 ALIGN=MIDDLE  ALT=\"3x%5E2+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the slope of the tangent line at the point of tangency should be m = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%5E2+-+2 ALIGN=MIDDLE  ALT=\"3a%5E2+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%5E2+-+2+=++%28b-3%29%2F%28a%2B1%29 ALIGN=MIDDLE  ALT=\"3a%5E2+-+2+=++%28b-3%29%2F%28a%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283a%5E2+-+2%29%28a%2B1%29+=++b-3 ALIGN=MIDDLE  ALT=\"%283a%5E2+-+2%29%28a%2B1%29+=++b-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <==>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283a%5E2+-+2%29%28a%2B1%29+=++a%5E3+-+2a+%2B1-3 ALIGN=MIDDLE  ALT=\"%283a%5E2+-+2%29%28a%2B1%29+=++a%5E3+-+2a+%2B1-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "since (a,b) is a point on the curve.\r<BR>\n" );
document.write( "\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E3%2B3a%5E2+=+0 ALIGN=MIDDLE  ALT=\"2a%5E3%2B3a%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  after simplification.\r<BR>\n" );
document.write( "\n" );
document.write( "<==><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%282a%2B3%29+=+0 ALIGN=MIDDLE  ALT=\"a%5E2%282a%2B3%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ==> a = 0 or a = -3/2.\r<BR>\n" );
document.write( "\n" );
document.write( "==> b = 1 or b = 5/8, respectively, after substitution.\r<BR>\n" );
document.write( "\n" );
document.write( "Thus the two points of tangency are (0,1) and (-3/2,5/8).\r<BR>\n" );
document.write( "\n" );
document.write( "==> There are two tangent lines.\r<BR>\n" );
document.write( "\n" );
document.write( "The tangent line passing through (0,1) and (-1,3) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y+=+-2x+%2B1%29 ALIGN=MIDDLE  ALT=\"highlight%28y+=+-2x+%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "And, the tangent line passing through (-3/2,5/8) and (-1,3) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%284y+=+19x+%2B31%29 ALIGN=MIDDLE  ALT=\"highlight%284y+=+19x+%2B31%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );