document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=556161>Question 556161</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 	<BR>\n" );
document.write( "Find a quadratic model for each set of values.\r<BR>\n" );
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document.write( "(-1, 1), (1, 1), (3, 9)<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 #362071 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=362071\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Two points determine a linear function; three points determine a quadratic function.<BR>\n" );
document.write( "A quadratic function can be written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <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;\" ><BR>\n" );
document.write( "In general, you can always substitute the coordinates of each of 3 points to get 3 equations.<BR>\n" );
document.write( "For your problem,<BR>\n" );
document.write( "(-1,1) ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=a%28-1%29%5E2%2Bb%28-1%29%2Bc ALIGN=MIDDLE  ALT=\"1=a%28-1%29%5E2%2Bb%28-1%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=a-b%2Bc ALIGN=MIDDLE  ALT=\"1=a-b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(1,1) ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=a%281%29%5E2%2Bb%281%29%2Bc ALIGN=MIDDLE  ALT=\"1=a%281%29%5E2%2Bb%281%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=a%2Bb%2Bc ALIGN=MIDDLE  ALT=\"1=a%2Bb%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(3,9) ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=a%283%29%5E2%2Bb%283%29%2Bc ALIGN=MIDDLE  ALT=\"9=a%283%29%5E2%2Bb%283%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=9a%2B3b%2Bc ALIGN=MIDDLE  ALT=\"9=9a%2B3b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You got a system of equations.<BR>\n" );
document.write( "From there, you solve the system for a, b, and c and those coefficients determine your quadratic function.<BR>\n" );
document.write( "In your case, symmetrical points (-1, 1), and (1, 1) tell you that the axis of symmetry will be the y-axis (the line x=0), making b=0.<BR>\n" );
document.write( "In your case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The simplest quadratic function (the mother of all quadratic functions) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Without grabbing your pencil (or pen), you can see that it passes through all 3 of your points. There is only one quadratic function that passes through any set of 3 points, so<BR>\n" );
document.write( "Your function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "(But you can solve the system of equations if it makes you, or your teacher happy. It's an easy one.) </SPAN>\n" );
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