document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=350591>Question 350591</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that A, B, C, and D are constants and f is the cubic polynomial f(x)=Ax³+Bx²+Cx+D. Suppose also that the tangent line to y = f(x) at x = 0 is y = x and the tangent line at x = 2 is given by y = 2x – 3. Find the values of A, B, C, and D. Then sketch the graph of y = f(x) and the two tangent lines for -2 &#8804; x &#8804; 4. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #250583 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=250583\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=Ax%5E3%2BBx%5E2%2BCx%2BD ALIGN=MIDDLE  ALT=\"y=Ax%5E3%2BBx%5E2%2BCx%2BD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The slope of the tangent line at a given point is the value of the derivative at that point. <BR>\n" );
document.write( "The derivative of the function is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=3Ax%5E2%2B2Bx%2BC ALIGN=MIDDLE  ALT=\"dy%2Fdx=3Ax%5E2%2B2Bx%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At x=0, the slope of the tangent line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1 ALIGN=MIDDLE  ALT=\"m=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=dy%2Fdx%280%29=3A%280%29%2B2B%280%29%2BC=1 ALIGN=MIDDLE  ALT=\"m=dy%2Fdx%280%29=3A%280%29%2B2B%280%29%2BC=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28C=1%29 ALIGN=MIDDLE  ALT=\"highlight%28C=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "At x=2, the slope of the tangent line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=2 ALIGN=MIDDLE  ALT=\"m=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=dy%2Fdx%282%29=3A%282%29%5E2%2B2B%282%29%2B1=2 ALIGN=MIDDLE  ALT=\"m=dy%2Fdx%282%29=3A%282%29%5E2%2B2B%282%29%2B1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12A%2B4B%2B1=2 ALIGN=MIDDLE  ALT=\"12A%2B4B%2B1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "1.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12A%2B4B=1 ALIGN=MIDDLE  ALT=\"12A%2B4B=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Using the tangent line at x=0, you also know the intersection point because,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x ALIGN=MIDDLE  ALT=\"y=x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(0,0) is also a point on the cubic.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=Ax%5E3%2BBx%5E2%2BCx%2BD ALIGN=MIDDLE  ALT=\"y=Ax%5E3%2BBx%5E2%2BCx%2BD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=A%280%29%2BB%280%29%2B%280%29%2BD=0 ALIGN=MIDDLE  ALT=\"y=A%280%29%2BB%280%29%2B%280%29%2BD=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28D=0%29 ALIGN=MIDDLE  ALT=\"highlight%28D=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Similarly at x=2,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x-3 ALIGN=MIDDLE  ALT=\"y=2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%282%29-3 ALIGN=MIDDLE  ALT=\"y=2%282%29-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=Ax%5E3%2BBx%5E2%2Bx ALIGN=MIDDLE  ALT=\"y=Ax%5E3%2BBx%5E2%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=A%282%29%5E3%2BB%282%29%5E2%2B2 ALIGN=MIDDLE  ALT=\"1=A%282%29%5E3%2BB%282%29%5E2%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=8A%2B4B%2B2 ALIGN=MIDDLE  ALT=\"1=8A%2B4B%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8A%2B4B=-1 ALIGN=MIDDLE  ALT=\"8A%2B4B=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtract eq. 2 from eq. 1,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12A%2B4B-8A-4B=1-%28-1%29 ALIGN=MIDDLE  ALT=\"12A%2B4B-8A-4B=1-%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4A=2 ALIGN=MIDDLE  ALT=\"4A=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A=1%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%28A=1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then from eq. 2,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%281%2F2%29%2B4B=-1 ALIGN=MIDDLE  ALT=\"8%281%2F2%29%2B4B=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B4B=-1 ALIGN=MIDDLE  ALT=\"4%2B4B=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4B=-5 ALIGN=MIDDLE  ALT=\"4B=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28B=-5%2F4%29 ALIGN=MIDDLE  ALT=\"highlight%28B=-5%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3%2C5%2C-10%2C10%2Ccircle%280%2C0%2C0.15%29%2Ccircle%282%2C1%2C0.15%29%2Cgraph%28300%2C300%2C-3%2C5%2C-10%2C10%2C%281%2F2%29x%5E3-%285%2F4%29x%5E2%2Bx%2Cx%2C2x-3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );