document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=178109>Question 178109</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a quadratic function in standard form for each set of points.<BR>\n" );
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document.write( "(0, 3), (1, –4), (2, –9)\r<BR>\n" );
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document.write( "(0 , –4), (1, 0), (2, 2)  \r<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 #133061 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=133061\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "If a quadratic function <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x) = ax^2 + bx + c\"> has a graph containing a point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_1,y_1)\"> then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ax_1^2 + bx_1 + c = y_1\">.\r<BR>\n" );
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document.write( "So, since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x) = ax^2 + bx + c\"> contains the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (0, 3)\"> we know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a(0)^2 + b(0) + c = 3\"> which is to say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c = 3\">.\r<BR>\n" );
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document.write( "Similarly, since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x) = ax^2 + bx + c\"> contains the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (1, -4)\"> we know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a(1)^2 + b(1) + c = -4\">.  But since we already know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c = 3\">, we can say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a + b + 3 = -4\"> which is to say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a + b = -7\">.\r<BR>\n" );
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document.write( "Continuing the process, the third point, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (2, -9)\">, will result in: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a(2)^2 + b(2) + 3 = -9\"> which is to say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 4a + 2b = -12\">.\r<BR>\n" );
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document.write( "Now:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {         } \math a + b = -7\"> and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {         } \math 4a + 2b = -12\">\r<BR>\n" );
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document.write( "form a system of two linear equations in two variables.  Given that the coefficents on both variables in the first equation is 1, this system lends itself well to solution by substitution.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a + b = -7 \text { } \Rightarrow \text{ }\math a = -b - 7\">\r<BR>\n" );
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document.write( "Substituting this expression for <i>a</i> into the second equation yields:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 4(-b-7) + 2b = -12 \text { } \Rightarrow \text{ }\math  -4b -28 + 2b = -12 \text { } \Rightarrow \text{ }\math -2b = 16 \text { } \Rightarrow \text{ }\math b = -8\">\r<BR>\n" );
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document.write( "Now that we know <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b = -8\"> we can substitute this value into <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a = -b - 7 \text { } \Rightarrow \text{ }\math a = -(-8) - 7 = 1\">\r<BR>\n" );
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document.write( "Now that we have values for <i>a</i>, <i>b</i>, and <i>c</i>, we can write the quadratic function specifically:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x) = ax^2 + bx + c = x^2 - 8x + 3\">    \r<BR>\n" );
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document.write( "Do the other problem the same way.<BR>\n" );
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