document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=176546>Question 176546</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0395937787/algebrahomeworkh>algebra 2</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> y=x^2-2x-1 graph the quadratic function. label the vertex and axis of semitry. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #131631 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=131631\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
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document.write( "Table of Contents:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"#vertex\">Step 1:</a> Finding the Vertex<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"#left_points\">Step 2:</a> Finding two points to left of axis of symmetry<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"#right_points\">Step 3:</a> Reflecting two points to get points right of axis of symmetry<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"#plot_points\">Step 4:</a> Plotting the Points (with table)<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"#graph_parabola\">Step 5:</a> Graphing the Parabola\r<BR>\n" );
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document.write( "In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-2x-1 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-2x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can follow the steps:\r<BR>\n" );
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document.write( "Step 1) Find the vertex (the vertex is the either the highest or lowest point on the graph). Also, the vertex is at the axis of symmetry of the parabola (ie it divides it in two).\r<BR>\n" );
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document.write( "Step 2) Once you have the vertex, find two points on the left side of the axis of symmetry (the line that vertically runs through the vertex).\r<BR>\n" );
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document.write( "Step 3) Reflect those two points over the axis of symmetry to get two more points on the right side of the axis of symmetry.\r<BR>\n" );
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document.write( "Step 4) Plot all of the points found (including the vertex).\r<BR>\n" );
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document.write( "Step 5) Draw a curve through all of the points to graph the parabola.\r<BR>\n" );
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document.write( "Let's go through these steps in detail\r<BR>\n" );
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document.write( "Step 1) <h4>Finding the vertex:</h4>\r<BR>\n" );
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document.write( "In order to find the vertex, we first need to find the x-coordinate of the vertex.\r<BR>\n" );
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document.write( "To find the x-coordinate of the vertex, use this formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F%282a%29\"   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=x=%28-b%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given formula.\r<BR>\n" );
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document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-2x-1 ALIGN=MIDDLE  ALT=\"y=x%5E2-2x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-1 ALIGN=MIDDLE  ALT=\"c=-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=x=%28-%28-2%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x=%28-%28-2%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   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=x=%282%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x=%282%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <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=2 ALIGN=MIDDLE  ALT=\"2\"   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=x=%282%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x=%282%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and <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=2 ALIGN=MIDDLE  ALT=\"2\"   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=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide.\r<BR>\n" );
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document.write( "So the x-coordinate of the vertex is <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;\" >. Note: this means that the axis of symmetry is also <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;\" >.\r<BR>\n" );
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document.write( "Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=x%5E2-2x-1 ALIGN=MIDDLE  ALT=\"f%281%29=x%5E2-2x-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=f%281%29=%281%29%5E2-2%281%29-1 ALIGN=MIDDLE  ALT=\"f%281%29=%281%29%5E2-2%281%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=1%281%29-2%281%29-1 ALIGN=MIDDLE  ALT=\"f%281%29=1%281%29-2%281%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" >.\r<BR>\n" );
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document.write( "So the y-coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2 ALIGN=MIDDLE  ALT=\"y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So the vertex is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(1,-2\right)\">.\r<BR>\n" );
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document.write( "---------------------------------------------------------------------\r<BR>\n" );
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document.write( "Step 2) <h4>Find two points to the left of the axis of symmetry:</h4>\r<BR>\n" );
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document.write( "Let's find the y value when <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;\" >\r<BR>\n" );
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document.write( "So the first point to the left of the axis of symmetry is (-1,2)\r<BR>\n" );
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document.write( "Let's find the y value when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   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=f%280%29=0%2B0-1 ALIGN=MIDDLE  ALT=\"f%280%29=0%2B0-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   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=f%280%29=-1 ALIGN=MIDDLE  ALT=\"f%280%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So the second point to the left of the axis of symmetry is (0,-1)\r<BR>\n" );
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document.write( "Step 3) <h4>Reflecting the two points over the axis of symmetry:</h4>\r<BR>\n" );
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document.write( "Now remember, the parabola is symmetrical about the axis of symmetry (which is <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;\" >)\r<BR>\n" );
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document.write( "This means the y-value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is one unit from the axis of symmetry) is equal to the y-value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is also one unit from the axis of symmetry). So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which gives us the point (2,-1). So we essentially reflected the point (0,-1) over to (2,-1).\r<BR>\n" );
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document.write( "Also, the y-value for <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;\" > (which is two units from the axis of symmetry) is equal to the y-value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is also two units from the axis of symmetry). So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which gives us the point (3,2). So we essentially reflected the point (-1,2) over to (3,2).\r<BR>\n" );
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document.write( "<a name=\"plot_points\">\r<BR>\n" );
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document.write( "Step 4) <h4>Plotting the points:</h4>\r<BR>\n" );
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document.write( "Now lets make a table of the values we have calculated:\r<BR>\n" );
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document.write( "  <TABLE  border=\"1\" width=\"250\">\n" );
document.write( "<th>x</th><th>y</th><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">2</TD></tr><tr><TD  align=\"center\">0</TD><TD  align=\"center\">-1</TD></tr><tr><TD  align=\"center\">1</TD><TD  align=\"center\">-2</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">-1</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">2</TD></tr>\n" );
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document.write( "Now let's plot the points:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C+0%29%2C%0D%0Acircle%28-1%2C2%2C0.05%29%2C%0D%0Acircle%28-1%2C2%2C0.08%29%2C%0D%0Acircle%28-1%2C2%2C0.10%29%2C%0D%0Acircle%280%2C-1%2C0.05%29%2C%0D%0Acircle%280%2C-1%2C0.08%29%2C%0D%0Acircle%280%2C-1%2C0.10%29%2C%0D%0Acircle%281%2C-2%2C0.05%29%2C%0D%0Acircle%281%2C-2%2C0.08%29%2C%0D%0Acircle%281%2C-2%2C0.10%29%2C%0D%0Acircle%282%2C-1%2C0.05%29%2C%0D%0Acircle%282%2C-1%2C0.08%29%2C%0D%0Acircle%282%2C-1%2C0.10%29%2C%0D%0Acircle%283%2C2%2C0.05%29%2C%0D%0Acircle%283%2C2%2C0.08%29%2C%0D%0Acircle%283%2C2%2C0.10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 5) <h4>Drawing a curve through all of the points:</h4>\r<BR>\n" );
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document.write( "Now draw a curve through all of the points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-2x-1 ALIGN=MIDDLE  ALT=\"y=x%5E2-2x-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=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C+x%5E2-2x-1%29%2C%0D%0Acircle%28-1%2C2%2C0.05%29%2C%0D%0Acircle%28-1%2C2%2C0.08%29%2C%0D%0Acircle%28-1%2C2%2C0.10%29%2C%0D%0Acircle%280%2C-1%2C0.05%29%2C%0D%0Acircle%280%2C-1%2C0.08%29%2C%0D%0Acircle%280%2C-1%2C0.10%29%2C%0D%0Acircle%281%2C-2%2C0.05%29%2C%0D%0Acircle%281%2C-2%2C0.08%29%2C%0D%0Acircle%281%2C-2%2C0.10%29%2C%0D%0Acircle%282%2C-1%2C0.05%29%2C%0D%0Acircle%282%2C-1%2C0.08%29%2C%0D%0Acircle%282%2C-1%2C0.10%29%2C%0D%0Acircle%283%2C2%2C0.05%29%2C%0D%0Acircle%283%2C2%2C0.08%29%2C%0D%0Acircle%283%2C2%2C0.10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-2x-1 ALIGN=MIDDLE  ALT=\"y=x%5E2-2x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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