document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=982609>Question 982609</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are two tangent line to the curve y=4x-x^2 that pass through the point (2,5). find the equations of these two lines and make a sketch to verify your results </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #603436 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39618)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=603436\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Slope of the tangency line must be same as dy/dx for the parabola, and this dy/dx will be a variable expression. \r<BR>\n" );
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document.write( "This slope based on the parabola:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dy%2Fdx%29%284x-x%5E2%29=4-2x ALIGN=MIDDLE  ALT=\"%28dy%2Fdx%29%284x-x%5E2%29=4-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Again, this is a SLOPE for the as yet unknown tangent lines.\r<BR>\n" );
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document.write( "Here is a look just at the parabola:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-6%2C6%2C-6%2C6%2C4x-x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-6%2C6%2C-6%2C6%2C4x-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You should be able to understand that (2,5) is on the axis of symmetry and is 1 unit above the vertex.\r<BR>\n" );
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document.write( "The tangent line built using Point-Slope equation form:<BR>\n" );
document.write( "This must contain the given point (2,5);<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=%284-2x%29%28x-2%29 ALIGN=MIDDLE  ALT=\"y-5=%284-2x%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and doing the algebra steps....<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28y=-2x%5E2%2B8x-3%29 ALIGN=MIDDLE  ALT=\"highlight_green%28y=-2x%5E2%2B8x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >----Understand this without its appearance fooling you.  This represents a LINE, but it is variable.\r<BR>\n" );
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document.write( "If you sketch the graph, the understanding should intuitively be this:<BR>\n" );
document.write( "We must have  (tangent line) - (parabola) = 0, in order for the intersection to be only any single point.<BR>\n" );
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document.write( "Use the formulas of y for each of those objects.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2%2B8x-3-%284x-x%5E2%29=0 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B8x-3-%284x-x%5E2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Do the algebra steps...<BR>\n" );
document.write( "omitting them here...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28highlight_green%28x%5E2-4x%2B3=0%29%29 ALIGN=MIDDLE  ALT=\"highlight_green%28highlight_green%28x%5E2-4x%2B3=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >----------This will give the x coordinates of the two tangent lines at the points of tangency on the parabola.<BR>\n" );
document.write( "... and it is FACTORABLE.\r<BR>\n" );
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document.write( "x=1  or  x=3\r<BR>\n" );
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document.write( "Use each individually in the line equation  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=%284-2x%29%28x-2%29 ALIGN=MIDDLE  ALT=\"y-5=%284-2x%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is equation for line intersection the parabola.  ONLY substitute for the slope, which is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-2x ALIGN=MIDDLE  ALT=\"4-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >; so that you can get the equation of the TANGENT line.<BR>\n" );
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document.write( "Doing that and simplifying :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%284-2%2A1%29%28x-2%29%2B5 ALIGN=MIDDLE  ALT=\"y=%284-2%2A1%29%28x-2%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%284-2%2A3%29%28x-2%29%2B5 ALIGN=MIDDLE  ALT=\"y=%284-2%2A3%29%28x-2%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=-2x%2B9%29 ALIGN=MIDDLE  ALT=\"highlight%28y=-2x%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The two tangent lines.<BR>\n" );
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