document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=292650>Question 292650</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What are the coordinates of the turning point for the graph of the parabola whose equation is y=2x^2+4x-7? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #211346 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=211346\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We essentially want to convert the equation into vertex form. To do that, we need to complete the square. Note: the vertex of the parabola is the turning point.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B4x-7 ALIGN=MIDDLE  ALT=\"2x%5E2%2B4x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%5E2%2B2x-7%2F2%29 ALIGN=MIDDLE  ALT=\"2%28x%5E2%2B2x-7%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This step is very important: the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <font size=4><b>must</b></font> be equal to 1.\r<BR>\n" );
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document.write( "Take half of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%282%29=1 ALIGN=MIDDLE  ALT=\"%281%2F2%29%282%29=1\"   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 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%5E2=%281%29%281%29=1 ALIGN=MIDDLE  ALT=\"%281%29%5E2=%281%29%281%29=1\"   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=2%28x%5E2%2B2x%2Bhighlight%281-1%29-7%2F2%29 ALIGN=MIDDLE  ALT=\"2%28x%5E2%2B2x%2Bhighlight%281-1%29-7%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <font size=4><b>and</b></font> subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > inside the parenthesis. Make sure to place this after the \"x\" term. Notice how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-1=0 ALIGN=MIDDLE  ALT=\"1-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the expression is not changed.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28%28x%5E2%2B2x%2B1%29-1-7%2F2%29 ALIGN=MIDDLE  ALT=\"2%28%28x%5E2%2B2x%2B1%29-1-7%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group the first three terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28%28x%2B1%29%5E2-1-7%2F2%29 ALIGN=MIDDLE  ALT=\"2%28%28x%2B1%29%5E2-1-7%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x%2B1 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B1%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=2%28%28x%2B1%29%5E2-9%2F2%29 ALIGN=MIDDLE  ALT=\"2%28%28x%2B1%29%5E2-9%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%2B1%29%5E2%2B2%28-9%2F2%29 ALIGN=MIDDLE  ALT=\"2%28x%2B1%29%5E2%2B2%28-9%2F2%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=2%28x%2B1%29%5E2-9 ALIGN=MIDDLE  ALT=\"2%28x%2B1%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
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document.write( "So after completing the square, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B4x-7 ALIGN=MIDDLE  ALT=\"2x%5E2%2B4x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > transforms to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%2B1%29%5E2-9 ALIGN=MIDDLE  ALT=\"2%28x%2B1%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B4x-7=2%28x%2B1%29%5E2-9 ALIGN=MIDDLE  ALT=\"2x%5E2%2B4x-7=2%28x%2B1%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B4x-7=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B4x-7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%2B1%29%5E2-9=0 ALIGN=MIDDLE  ALT=\"2%28x%2B1%29%5E2-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%5E2%2B4x-7 ALIGN=MIDDLE  ALT=\"y=2x%5E2%2B4x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%28x%2B1%29%5E2-9 ALIGN=MIDDLE  ALT=\"y=2%28x%2B1%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%28x%2B1%29%5E2-9 ALIGN=MIDDLE  ALT=\"y=2%28x%2B1%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > is now 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=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-1 ALIGN=MIDDLE  ALT=\"h=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-9 ALIGN=MIDDLE  ALT=\"k=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Remember, the vertex of <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;\" > is (h,k).\r<BR>\n" );
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document.write( "So the vertex of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%28x%2B1%29%5E2-9 ALIGN=MIDDLE  ALT=\"y=2%28x%2B1%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (-1,-9). \r<BR>\n" );
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document.write( "This consequently means that the vertex of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%5E2%2B4x-7 ALIGN=MIDDLE  ALT=\"y=2x%5E2%2B4x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (-1,-9) (since the two equations are equivalent) \r<BR>\n" );
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document.write( "So the turning point of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%5E2%2B4x-7 ALIGN=MIDDLE  ALT=\"y=2x%5E2%2B4x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (-1,-9)\r<BR>\n" );
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