document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=224743>Question 224743</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Complete each of the following for the function g(x)=-3x^2-6x+1<BR>\n" );
document.write( "a.Find the vertex of g(x)<BR>\n" );
document.write( "b.Find the x-intercept <BR>\n" );
document.write( "c.Find the y-intercept  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #167908 by </B><A HREF=/tutors/aboutme.mpl?userid=drj><B>drj(1380)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drj><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=drj><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=167908\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Complete each of the following for the function g(x)=-3x^2-6x+1\r<BR>\n" );
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document.write( "a.Find the vertex of g(x)\r<BR>\n" );
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document.write( "b.Find the x-intercept \r<BR>\n" );
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document.write( "c.Find the y-intercept \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=-3x%5E2-6x%2B1 ALIGN=MIDDLE  ALT=\"g%28x%29=-3x%5E2-6x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 1. The general formula for a parabola is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-b=c%28x-a%29%5E2 ALIGN=MIDDLE  ALT=\"y-b=c%28x-a%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where x=a and y=b is the vertex or at point (a,b).  Here, c is a constant.\r<BR>\n" );
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document.write( "Step 2. Put <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=y=-3x%5E2-6x%2B1 ALIGN=MIDDLE  ALT=\"g%28x%29=y=-3x%5E2-6x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > in  standard form by completing the square in x with the following steps.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-3x%5E2-6x%2B1=-3x%5E2-6x-3%2B3%2B1=3%28x%2B1%29%2B4 ALIGN=MIDDLE  ALT=\"y=-3x%5E2-6x%2B1=-3x%5E2-6x-3%2B3%2B1=3%28x%2B1%29%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  where we added zero 0=-3+3 to complete the square\r<BR>\n" );
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document.write( "Factor out -3 in 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=y=-3%28x%5E2%2B2x%2B1%29%2B3%2B1=-3%28x%2B1%29%5E2%2B4 ALIGN=MIDDLE  ALT=\"y=-3%28x%5E2%2B2x%2B1%29%2B3%2B1=-3%28x%2B1%29%5E2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Subtract 4 from both sides of the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-4=-3%28x%2B1%29%5E2%2B4-4 ALIGN=MIDDLE  ALT=\"y-4=-3%28x%2B1%29%5E2%2B4-4\"   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-4=-3%28x%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"y-4=-3%28x%2B1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "ANSWER:  So the vertex is located at point (-1, 4)\r<BR>\n" );
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document.write( "Step 3.  To find the x-intercept,  set g(x)=y=0.  We use the quadratic formula given as\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "where a=-3, b=-6 and c=1\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2B-6x%2B1+=+0 ALIGN=MIDDLE  ALT=\"-3x%5E2%2B-6x%2B1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
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document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-6%29%5E2-4%2A-3%2A1=48 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-6%29%5E2-4%2A-3%2A1=48\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  Discriminant d=48 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--6%2B-sqrt%28+48+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--6%2B-sqrt%28+48+%29%29%2F2%5Ca\"   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=x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+48+%29%29%2F2%5C-3+=+-2.15470053837925 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+48+%29%29%2F2%5C-3+=+-2.15470053837925\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-6%29-sqrt%28+48+%29%29%2F2%5C-3+=+0.154700538379251 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-6%29-sqrt%28+48+%29%29%2F2%5C-3+=+0.154700538379251\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2B-6x%2B1 ALIGN=MIDDLE  ALT=\"-3x%5E2%2B-6x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2B-6x%2B1+=+-3%28x--2.15470053837925%29%2A%28x-0.154700538379251%29 ALIGN=MIDDLE  ALT=\"-3x%5E2%2B-6x%2B1+=+-3%28x--2.15470053837925%29%2A%28x-0.154700538379251%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -2.15470053837925, 0.154700538379251.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-3%2Ax%5E2%2B-6%2Ax%2B1+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-3%2Ax%5E2%2B-6%2Ax%2B1+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Step 4.  To find the y-intercept set x=0 in g(x)=y=-3x^2-6x+1 \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%280%29=y=-3%2A0%5E2-6%2A0%2B1 ALIGN=MIDDLE  ALT=\"g%280%29=y=-3%2A0%5E2-6%2A0%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "or y=1.   \r<BR>\n" );
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document.write( "The y-intercept is at (0,1) which is consistent with the above graph.\r<BR>\n" );
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document.write( "I hope the above steps and explanation were helpful.\r<BR>\n" );
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document.write( "For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.\r<BR>\n" );
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document.write( "And good luck in your studies!\r<BR>\n" );
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document.write( "Respectfully,<BR>\n" );
document.write( "Dr J\r<BR>\n" );
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document.write( "http://www.FreedomUniversity.TV\r<BR>\n" );
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document.write( " </SPAN>\n" );
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