document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=622091>Question 622091</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to learn this with the reinforcement of detailed instructions on how to get to these final results. Thank you.\r<BR>\n" );
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document.write( "The monthly value of a share of ACME Corporation, in dollars, since January 2009 are shown in the table below. The values of the table below model the behavior of a quadratic function when it is graphed.\r<BR>\n" );
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document.write( "Month	------          Share Value (in dollars)<BR>\n" );
document.write( "0	------          55<BR>\n" );
document.write( "3	------          16<BR>\n" );
document.write( "5	------          0<BR>\n" );
document.write( "7	------          -8<BR>\n" );
document.write( "9	------          -8<BR>\n" );
document.write( "11	------          0<BR>\n" );
document.write( "13	------          16<BR>\n" );
document.write( "15	------          40\r<BR>\n" );
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document.write( "1.	Using the table above, find the quadratic equation that represents the monthly value of a share of ACME Corporation since January 2009. Use the variable x to represent the number of months after January 2009 and the variable y to represent the monthly value\r<BR>\n" );
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document.write( "2.	Find the y-intercept.  Interpret the meaning of the y-intercept in relation to the given real-life scenario\r<BR>\n" );
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document.write( "3.	Find the x-intercepts. Interpret the meaning of the x-intercepts in relation to the given real-life scenario.\r<BR>\n" );
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document.write( "4.	From the equation you found in problem 1, what is the concavity of the parabola? How do you know? What is the lowest or maximum monthly value of a share of ACME Corporation and when did that occur?\r<BR>\n" );
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document.write( "5.	Using the quadratic equation you found in problem 1, predict the value of a share of ACME Corporation in October 2010?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #391300 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=391300\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If this was a real life set of data, I would first plot the points to see what shape they suggest. Next, I would use some computer software to do linear regression and find the fit of the functions that I suspect best fit the data points.<BR>\n" );
document.write( "If I only had pencil and paper. I would hand-draw a smooth curve that fit the points, and use my knowledge of math to figure out an equation for the function that represents the data.<BR>\n" );
document.write( "The problem says it's a quadratic function, so we need to do none of that.<BR>\n" );
document.write( " <BR>\n" );
document.write( "1. Three points determine a quadratic function (we may have to solve a system of equations to find coefficients), and all points would be used to verify the fit.<BR>\n" );
document.write( "A quadratic function can be represented as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-k%292%2Bh ALIGN=MIDDLE  ALT=\"y=a%28x-k%292%2Bh\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=k ALIGN=MIDDLE  ALT=\"x=k\"   DISABLED_style= \"max-width:90%; height:auto;\" > being the axis of symmetry and (k,h) being the coordinates of the vertex.<BR>\n" );
document.write( "If y=0 for x=s and x=t, then quadratic polynomial can be factored, and the function could be represented as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-s%29%28x-t%29 ALIGN=MIDDLE  ALT=\"y=a%28x-s%29%28x-t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "We have that last option, and it may be the easiest way to solve the problem.<BR>\n" );
document.write( "We have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-11%29%28x-5%29 ALIGN=MIDDLE  ALT=\"y=a%28x-11%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we just have to find a.<BR>\n" );
document.write( "For x=0,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28-11%29%28-5%29=55a=55 ALIGN=MIDDLE  ALT=\"y=a%28-11%29%28-5%29=55a=55\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x-11%29%28x-5%29 ALIGN=MIDDLE  ALT=\"y=%28x-11%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-16x%2B55 ALIGN=MIDDLE  ALT=\"y=x%5E2-16x%2B55\"   DISABLED_style= \"max-width:90%; height:auto;\" > could be the answer.<BR>\n" );
document.write( "We may want to express it in another way,<BR>\n" );
document.write( "completing the square for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-16x ALIGN=MIDDLE  ALT=\"x%5E2-16x\"   DISABLED_style= \"max-width:90%; height:auto;\" > like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-55=x%5E2-16x ALIGN=MIDDLE  ALT=\"y-55=x%5E2-16x\"   DISABLED_style= \"max-width:90%; height:auto;\" > -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-55%2B64=x%5E2-16x%2B64 ALIGN=MIDDLE  ALT=\"y-55%2B64=x%5E2-16x%2B64\"   DISABLED_style= \"max-width:90%; height:auto;\" > -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B9=%28x-8%29%5E2 ALIGN=MIDDLE  ALT=\"y%2B9=%28x-8%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=%28x-8%29%5E2-9%29 ALIGN=MIDDLE  ALT=\"highlight%28y=%28x-8%29%5E2-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The second option works well too, because the data suggests that the axis of symmetry is x=8, so we expect the equation to be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-8%29%5E2%2Bh ALIGN=MIDDLE  ALT=\"y=a%28x-8%29%5E2%2Bh\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "With the (x,y) coordinates of a couple of points we could set up a system of equations and find a and h.<BR>\n" );
document.write( "For x=9, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%289-8%29%5E2%2Bh=a%2Bh=-8 ALIGN=MIDDLE  ALT=\"y=a%289-8%29%5E2%2Bh=a%2Bh=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "For x=11, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%2811-8%29%5E2%2Bh=a%2A3%5E2%2Bh=9a%2Bh=0 ALIGN=MIDDLE  ALT=\"y=a%2811-8%29%5E2%2Bh=a%2A3%5E2%2Bh=9a%2Bh=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "give us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bh=-8%2C9a%2Bh=0%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bh=-8%2C9a%2Bh=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which gives us a=8 and h=-9,<BR>\n" );
document.write( "so we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=%28x-8%29%5E2-9%29 ALIGN=MIDDLE  ALT=\"highlight%28y=%28x-8%29%5E2-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2. The y-intercept is the y value when x=0. It is y=55. It is the share value for x=0, month zero, January 2009. <BR>\n" );
document.write( "3. The x-intercepts are the x values where y=0, x=7 and x=9. The share value was zero at months 7 and 9 after January 2009. That happened on August and October 2009. <BR>\n" );
document.write( "4. The equation has a negative coefficient a, so the parabola opens up. It's concave upwards, and the graph looks like a smile. The lowest point is the vertex (8,-9). It represents the lowest share value (-0), which occurred at month 8, in September 2009. The value of a share of ACME Corporation was -$9 then. </SPAN>\n" );
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