document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=127805>Question 127805</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I'm having trouble with the following function. Can I please have some help solving?\r<BR>\n" );
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document.write( "A quadratic function is given.<BR>\n" );
document.write( "f(x) = 10x^2 +x - 1<BR>\n" );
document.write( "(a) Express the quadratic function in standard form.<BR>\n" );
document.write( "f(x) = a(x - h)^2 + k where<BR>\n" );
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document.write( "h = <BR>\n" );
document.write( "k = \r<BR>\n" );
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document.write( "(c) Find its minimum or maximum value. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #93667 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=93667\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10+x%5E2%2B1+x-1 ALIGN=MIDDLE  ALT=\"y=10+x%5E2%2B1+x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B1=10+x%5E2%2B1+x ALIGN=MIDDLE  ALT=\"y%2B1=10+x%5E2%2B1+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B1=10%28x%5E2%2B%281%2F10%29x%29 ALIGN=MIDDLE  ALT=\"y%2B1=10%28x%5E2%2B%281%2F10%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the leading coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Take half of the x coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F10 ALIGN=MIDDLE  ALT=\"1%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F20 ALIGN=MIDDLE  ALT=\"1%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%281%2F10%29=1%2F20 ALIGN=MIDDLE  ALT=\"%281%2F2%29%281%2F10%29=1%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "Now square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F20 ALIGN=MIDDLE  ALT=\"1%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F400 ALIGN=MIDDLE  ALT=\"1%2F400\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F20%29%5E2=%281%2F20%29%281%2F20%29=1%2F400 ALIGN=MIDDLE  ALT=\"%281%2F20%29%5E2=%281%2F20%29%281%2F20%29=1%2F400\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B1=10%28x%5E2%2B%281%2F10%29x%2B1%2F400-1%2F400%29 ALIGN=MIDDLE  ALT=\"y%2B1=10%28x%5E2%2B%281%2F10%29x%2B1%2F400-1%2F400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F400 ALIGN=MIDDLE  ALT=\"1%2F400\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not change the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B1=10%28%28x%2B1%2F20%29%5E2-1%2F400%29 ALIGN=MIDDLE  ALT=\"y%2B1=10%28%28x%2B1%2F20%29%5E2-1%2F400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%281%2F10%29x%2B1%2F400 ALIGN=MIDDLE  ALT=\"x%5E2%2B%281%2F10%29x%2B1%2F400\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%2F20%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B1%2F20%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B1=10%28x%2B1%2F20%29%5E2-10%281%2F400%29 ALIGN=MIDDLE  ALT=\"y%2B1=10%28x%2B1%2F20%29%5E2-10%281%2F400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B1=10%28x%2B1%2F20%29%5E2-1%2F40 ALIGN=MIDDLE  ALT=\"y%2B1=10%28x%2B1%2F20%29%5E2-1%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10%28x%2B1%2F20%29%5E2-1%2F40-1 ALIGN=MIDDLE  ALT=\"y=10%28x%2B1%2F20%29%5E2-1%2F40-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B1 ALIGN=MIDDLE  ALT=\"%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides to isolate y\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10%28x%2B1%2F20%29%5E2-41%2F40 ALIGN=MIDDLE  ALT=\"y=10%28x%2B1%2F20%29%5E2-41%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
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document.write( "Now the quadratic is in vertex form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=10 ALIGN=MIDDLE  ALT=\"a=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-1%2F20 ALIGN=MIDDLE  ALT=\"h=-1%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-41%2F40 ALIGN=MIDDLE  ALT=\"k=-41%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember (h,k) is the vertex and \"a\" is the stretch/compression factor. Also \"a\" tells us which direction the parabola opens.\r<BR>\n" );
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document.write( "So in this case the vertex is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F20 ALIGN=MIDDLE  ALT=\"-1%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-41%2F40 ALIGN=MIDDLE  ALT=\"-41%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and the parabola opens upward  since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "Notice if we graph the original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10x%5E2%2B1x-1 ALIGN=MIDDLE  ALT=\"y=10x%5E2%2B1x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C10x%5E2%2B1x-1%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C10x%5E2%2B1x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10x%5E2%2B1x-1 ALIGN=MIDDLE  ALT=\"y=10x%5E2%2B1x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F20 ALIGN=MIDDLE  ALT=\"-1%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-41%2F40 ALIGN=MIDDLE  ALT=\"-41%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "Notice if we graph the final equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10%28x%2B1%2F20%29%5E2-41%2F40 ALIGN=MIDDLE  ALT=\"y=10%28x%2B1%2F20%29%5E2-41%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C10%28x%2B1%2F20%29%5E2-41%2F40%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C10%28x%2B1%2F20%29%5E2-41%2F40%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10%28x%2B1%2F20%29%5E2-41%2F40 ALIGN=MIDDLE  ALT=\"y=10%28x%2B1%2F20%29%5E2-41%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is also (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F20 ALIGN=MIDDLE  ALT=\"-1%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-41%2F40 ALIGN=MIDDLE  ALT=\"-41%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.\r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "The min or max occurs at the vertex and the max/min value is the y value of the vertex. So in this case the minimum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-41%2F40 ALIGN=MIDDLE  ALT=\"-41%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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