document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=461376>Question 461376</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Using the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-2x%2B1 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, find if it has a maximum or minimum and give that point. Also give x-intercepts. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #316436 by </B><A HREF=/tutors/aboutme.mpl?userid=math-vortex><B>math-vortex(648)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math-vortex><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=math-vortex><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=316436\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Using the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-2x%2B1 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, find if it has a maximum or minimum and give that point. Also give x-intercepts.\r<BR>\n" );
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document.write( "Find minimum/maximum point<BR>\n" );
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document.write( "This function represents a the standard form of an equation for a parabola, <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;\" >. <BR>\n" );
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document.write( "There is an easy way to tell if the parabola has a maximum or a minimum when the equation is in this form. If the coefficient of the x-squared term is positive the parabola opens upward, and the the function has a minimum at the vertex. If the coefficient is negative, the parabola opens downward, and the the function has a maximum value.<BR>\n" );
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document.write( "In your equation, the coefficient is 1 because you have <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;\" >. (Remember, if no coefficient is shown, it is understood to be 1.) Your parabola opens upward, and the function has a minimum value.<BR>\n" );
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document.write( "Parabolas have one maximum or minimum value located at the vertex. To find the vertex of your parabola, we use the formula, x = -b/2a. Looking at the equation, we see that b is -2 and a is 1, so x = -b/2a = -(-2)/(2(1)) = 1. To find the y-value of the vertex, substitute 1 for x in the equation. <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-2x%2B1=%281%29%5E2-2%281%29%2B1%29=0 ALIGN=MIDDLE  ALT=\"y=x%5E2-2x%2B1=%281%29%5E2-2%281%29%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Find the x-intercepts<BR>\n" );
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document.write( "The x-intercepts are the points where the parabola intersects the x-axis. To find the x-intercepts we set f(x) = 0. and solve for x.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-2x%2B1 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Set the function equal to zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=x%5E2-2x%2B1 ALIGN=MIDDLE  ALT=\"0=x%5E2-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Invert the order of the equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2B1=0 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is a quadratic equation. We have two choices for solving it: factoring, or the quadratic formula. In this case the equation looks pretty east to factor.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x-1%29=0 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that both factors are the same, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=0 ALIGN=MIDDLE  ALT=\"x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This tells us that when f(x) is 0, x is 1 and the x-intercept is the point (1,0).<BR>\n" );
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document.write( "A quick check using substitution, gives us<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-2x%2B1 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=%281%29%5E2-2%281%29%2B1 ALIGN=MIDDLE  ALT=\"0=%281%29%5E2-2%281%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=1-2%2B1 ALIGN=MIDDLE  ALT=\"0=1-2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=0 ALIGN=MIDDLE  ALT=\"0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (true)\r<BR>\n" );
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document.write( "Take a look at the graph:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2-2x%2B1%29 ALIGN=MIDDLE  ALT=\"graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2-2x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hope this helps!\r<BR>\n" );
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document.write( "Ms. Figgy<BR>\n" );
document.write( "math-in-the-vortex </SPAN>\n" );
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