document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=159802>Question 159802</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am trying to find the vertex for the following problem:\r<BR>\n" );
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document.write( "x^2+3x-4=0\r<BR>\n" );
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document.write( "All the help I could get would be appreciated.\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 #117838 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=117838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can find it in a couple of ways.<BR>\n" );
document.write( "the easiest is to remember that the vertex is given by the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+-b%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "how do you find this?<BR>\n" );
document.write( "your original equation is a quadratic equation.<BR>\n" );
document.write( "the general form of a quadratic equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c ALIGN=MIDDLE  ALT=\"y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "your equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2+%2B+3%2Ax+-+4 ALIGN=MIDDLE  ALT=\"y+=+x%5E2+%2B+3%2Ax+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "this is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+1%2Ax%5E2+%2B+3%2Ax+-+4 ALIGN=MIDDLE  ALT=\"y+=+1%2Ax%5E2+%2B+3%2Ax+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "if you compare your equation to the general form of the equation, you'll see that<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = 3<BR>\n" );
document.write( "c = -4<BR>\n" );
document.write( "plugging these values in into the equation for the vertex, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+-3%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which becomes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+-3%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which becomes<BR>\n" );
document.write( "x = -3/2<BR>\n" );
document.write( "------------------------------<BR>\n" );
document.write( "that finds the x value for the vertex.<BR>\n" );
document.write( "to find the y value for the vertex, plug the value of x for the vertex into the equation and solve.<BR>\n" );
document.write( "the equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2+%2B+3%2Ax+-+4 ALIGN=MIDDLE  ALT=\"y+=+x%5E2+%2B+3%2Ax+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-3%2F2%29%5E2+%2B+3%2A%28-3%2F2%29+-+4 ALIGN=MIDDLE  ALT=\"y+=+%28-3%2F2%29%5E2+%2B+3%2A%28-3%2F2%29+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2.25+%2B+%28-4.5%29+-+4 ALIGN=MIDDLE  ALT=\"y+=+2.25+%2B+%28-4.5%29+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-2.25+-+4 ALIGN=MIDDLE  ALT=\"y+=+-2.25+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which becomes y = -6.5<BR>\n" );
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document.write( "the vertex for this equation becomes (-3/2, -6.5).<BR>\n" );
document.write( "the x value is -3/2.<BR>\n" );
document.write( "the y value is -6.5.<BR>\n" );
document.write( "-----------------------------<BR>\n" );
document.write( "since the vertex is the point in the graph where the y value is either a maximum or a minimum, we can assume that this graph will change direction at that point.<BR>\n" );
document.write( "------------------------------<BR>\n" );
document.write( "outside of the equation for the vertex (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+-3%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >), the next best thing to know is whether the quadratic equation is pointing upwards (vertex is a maximum) or whether the quadratic equation is pointing downwards (vertex is a minimum).<BR>\n" );
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document.write( "going back to the general form of the equation (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c ALIGN=MIDDLE  ALT=\"y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >),<BR>\n" );
document.write( "the graph is pointing upwards (vertex is maximum value for y) when a is negative, and the graph is pointing downwards (vertex is minimum value for y) when a is positive.<BR>\n" );
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document.write( "this should be what you would think would happen intuitively.<BR>\n" );
document.write( "if y = x^2, then for negative values of x or positive values of x, y = x^2 will always be positive.  that tells you that the graph will be pointing downwards and the vertex will be a minimum.<BR>\n" );
document.write( "if y = -x^2, then for negative values of x or positive values of x, y = -x^2 will always be negative.  that tells you that the graph will be pointing upwards and the vertex will be a maximum.<BR>\n" );
document.write( "---------------------------------<BR>\n" );
document.write( "in the equation provided, a = 1 which is positive.<BR>\n" );
document.write( "we should expect then that the graph is pointing downwards and that the vertex of (-3/2,-6.5) is a minimum value.<BR>\n" );
document.write( "----------------------------------<BR>\n" );
document.write( "the graph of the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2+%2B+3%2Ax+-+4 ALIGN=MIDDLE  ALT=\"y+=+x%5E2+%2B+3%2Ax+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the graph looks like this.<BR>\n" );
document.write( "please scan below the graph for further comments.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-15%2C15%2C-10%2C100%2Cx%5E2%2B3%2Ax-4%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-15%2C15%2C-10%2C100%2Cx%5E2%2B3%2Ax-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "as you can see, the graph is point downwards and the vertex is the minimum.<BR>\n" );
document.write( "this is because a was positive (a*x^2 in the general form of the equation).  specifically, a was +1.<BR>\n" );
document.write( "you can also see that the minimum is where we calculated it to be.<BR>\n" );
document.write( "x = -3/2 is the same as -1.5.<BR>\n" );
document.write( "y = -6.25.<BR>\n" );
document.write( "vertex = (-1.5,-6.25)<BR>\n" );
document.write( "x value of the vertex was provided by the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+-b%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where a was 1, and b was 3.<BR>\n" );
document.write( "y value of the vertex was provided by plugging x value of -3/2 into the equation and solving.<BR>\n" );
document.write( "things to remember:<BR>\n" );
document.write( "general form of quadratic equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c ALIGN=MIDDLE  ALT=\"y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "formula for x-value of vertex is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+-b%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y-value of vertex is found by plugging x-value of vertex into the equation to be solved.<BR>\n" );
document.write( "when a is positive, equation points downward.<BR>\n" );
document.write( "when a is negative, equation points upward.<BR>\n" );
document.write( "when the equation points downward, the vertex is a minimum value of y.<BR>\n" );
document.write( "when the equation points upward, the vertex is a maximum value of y.\r<BR>\n" );
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