document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195006>Question 1195006</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would I go about sketching a graph for the total cost function, revenue function, and profit function:\r<BR>\n" );
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document.write( "C(x)=0.000002x^3-0.03x^2+400x+80,000\r<BR>\n" );
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document.write( "R(x)=-0.05x^2+600x\r<BR>\n" );
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document.write( "p(x)=-0.000002x^3-0.02x^2+200x-80,000 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #827341 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=827341\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I also recommend using technology like Desmos. <BR>\n" );
document.write( "GeoGebra is another useful tool you can use.\r<BR>\n" );
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document.write( "If your teacher wants you to \"plot by hand\", so to speak, then you'll need to plug in various x values to find corresponding y values. <BR>\n" );
document.write( "This way you'll get the (x,y) points that make up the curve.\r<BR>\n" );
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document.write( "For example, let's say we plugged in x = 10 into the C(x) function<BR>\n" );
document.write( "C(x)=0.000002x^3-0.03x^2+400x+80,000<BR>\n" );
document.write( "C(10)=0.000002(10)^3-0.03(10)^2+400(10)+80,000<BR>\n" );
document.write( "C(10)=83997.002\r<BR>\n" );
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document.write( "You could compute that by pencil/paper, but I recommend using a calculator to do the third step.\r<BR>\n" );
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document.write( "From here we can say that the point (10, 83997.002) is on the C(x) cubic curve.<BR>\n" );
document.write( "Repeat this process for other x values to generate as many points as needed. <BR>\n" );
document.write( "Then draw a curve through them all the best you can.\r<BR>\n" );
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document.write( "With regard to the revenue function R(x), we can factor out the GCF like so<BR>\n" );
document.write( "R(x)=-0.05x^2+600x<BR>\n" );
document.write( "R(x)=-0.05x(x-12000)\r<BR>\n" );
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document.write( "Solving R(x) = 0 has us get:<BR>\n" );
document.write( "-0.05x(x-12000) = 0<BR>\n" );
document.write( "-0.05x = 0 or x-12000 = 0<BR>\n" );
document.write( "x = 0 or x = 12000<BR>\n" );
document.write( "which are the two x intercepts.<BR>\n" );
document.write( "This is where the graph crosses the x axis.\r<BR>\n" );
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document.write( "Let's rewrite the revenue equation into vertex form<BR>\n" );
document.write( "Compare y = -0.05x^2+600x to y = ax^2+bx+c <BR>\n" );
document.write( "a = -0.05<BR>\n" );
document.write( "b = 600<BR>\n" );
document.write( "c = 0\r<BR>\n" );
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document.write( "The x coordinate of the vertex is <BR>\n" );
document.write( "h = -b/(2a)<BR>\n" );
document.write( "h = -600/(2(-0.05))<BR>\n" );
document.write( "h = 6000<BR>\n" );
document.write( "Or you could remark that the x coordinate of the vertex is the midpoint of the x intercepts<BR>\n" );
document.write( "h = (a+b)/2 = (0+12000)/2 = 6000<BR>\n" );
document.write( "This midpoint property is directly due to symmetry.\r<BR>\n" );
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document.write( "Then plug this into the original R(x) equation to find the y coordinate of the vertex<BR>\n" );
document.write( "y = -0.05x^2+600x<BR>\n" );
document.write( "y = -0.05(6000)^2+600(6000)<BR>\n" );
document.write( "y = 1,800,000\r<BR>\n" );
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document.write( "The vertex is located at (x,y) = (6000, 1800000)\r<BR>\n" );
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document.write( "The parabola opens downward because a < 0<BR>\n" );
document.write( "Think of \"negative 'a' value makes a negative frown\"<BR>\n" );
document.write( "Because the parabola opens downward, we know that the R(x) function maxes out at the vertex.<BR>\n" );
document.write( "This helps determine the max revenue which is what many managers solely care about.\r<BR>\n" );
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document.write( "The x intercepts and vertex should give a rough idea of how the parabola looks (in a general sense). <BR>\n" );
document.write( "Plug in various other x values to get more points to plot on the parabola.\r<BR>\n" );
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document.write( "Unfortunately the cubic curves aren't as easy to determine their traits about them. <BR>\n" );
document.write( "Often with many cubics, it's very difficult to determine the x intercepts by hand. This is why a graphing calculator is preferred.<BR>\n" );
document.write( "You could use calculus methods, but this might be beyond the scope of the course.<BR>\n" );
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