document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=193521>Question 193521</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780395977255/algebrahomeworkh>algebra and trigonometry structure and method book 2</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Find the vertex, focus, directrix, and axis of symmetry of each parabola. <BR>\n" );
document.write( "x^2+8y+4x-4=0<BR>\n" );
document.write( "We have been using the formulas y-k=a(x-h)^2 and x-h=a(y-k)^2 I don't understand which to use here, even after completing the square and its confusing because I don't know which is which. please explain and be specific. thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #145262 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=145262\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B8y%2B4x-4=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B8y%2B4x-4=0\"   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=x%5E2%2B4x%2B8y-4=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B4x%2B8y-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "Now we need to complete the square for the \"x\" terms.\r<BR>\n" );
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document.write( "Take half of the x coefficient 4 to get 2. Square 2 to get 4.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4x%2B4%2B8y-4=4 ALIGN=MIDDLE  ALT=\"x%5E2%2B4x%2B4%2B8y-4=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add the previous result 4 to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4x%2B4%2B8y-4-4=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B4x%2B4%2B8y-4-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 4 from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4x%2B4%2B8y-8=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B4x%2B4%2B8y-8=0\"   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=%28x%2B2%29%5E2%2B8y-8=0 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%5E2%2B8y-8=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4x%2B4 ALIGN=MIDDLE  ALT=\"x%5E2%2B4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B2%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=8y-8=-%28x%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"8y-8=-%28x%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%28y-1%29=-%28x%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"8%28y-1%29=-%28x%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF 8 from the left side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-1=-%281%2F8%29%28x%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"y-1=-%281%2F8%29%28x%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F8 ALIGN=MIDDLE  ALT=\"1%2F8\"   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-1=-%281%2F8%29%28x-%28-2%29%29%5E2 ALIGN=MIDDLE  ALT=\"y-1=-%281%2F8%29%28x-%28-2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2 ALIGN=MIDDLE  ALT=\"x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%28-2%29 ALIGN=MIDDLE  ALT=\"x-%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-1=-%281%2F8%29%28x-%28-2%29%29%5E2 ALIGN=MIDDLE  ALT=\"y-1=-%281%2F8%29%28x-%28-2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-k=a%28x-h%29%5E2 ALIGN=MIDDLE  ALT=\"y-k=a%28x-h%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-1%2F8 ALIGN=MIDDLE  ALT=\"a=-1%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-2 ALIGN=MIDDLE  ALT=\"h=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1 ALIGN=MIDDLE  ALT=\"k=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Vertex:\r<BR>\n" );
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document.write( "The vertex of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-k=a%28x-h%29%5E2 ALIGN=MIDDLE  ALT=\"y-k=a%28x-h%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (h,k). Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-2 ALIGN=MIDDLE  ALT=\"h=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1 ALIGN=MIDDLE  ALT=\"k=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the vertex of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-1=-%281%2F8%29%28x-%28-2%29%29%5E2 ALIGN=MIDDLE  ALT=\"y-1=-%281%2F8%29%28x-%28-2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (-2,1)\r<BR>\n" );
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document.write( "Focus: \r<BR>\n" );
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document.write( "First, we define <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=1%2F%284a%29 ALIGN=MIDDLE  ALT=\"p=1%2F%284a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the absolute value of \"p\" is the distance from the vertex to the focus. \r<BR>\n" );
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document.write( "Note: the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=1%2F%284a%29 ALIGN=MIDDLE  ALT=\"p=1%2F%284a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is derived from the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2=4p%28y-k%29 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2=4p%28y-k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The focus of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-k=a%28x-h%29%5E2 ALIGN=MIDDLE  ALT=\"y-k=a%28x-h%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(h,k+p\right)\">. In other words, simply add the value of \"p\" to the y coordinate of the vertex to get the focus.\r<BR>\n" );
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document.write( "So let's find \"p\": <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=1%2F%284a%29=1%2F%284%28-1%2F8%29%29=1%2F%28-1%2F2%29=-2 ALIGN=MIDDLE  ALT=\"p=1%2F%284a%29=1%2F%284%28-1%2F8%29%29=1%2F%28-1%2F2%29=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=-2 ALIGN=MIDDLE  ALT=\"p=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-2 ALIGN=MIDDLE  ALT=\"h=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1 ALIGN=MIDDLE  ALT=\"k=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=-2 ALIGN=MIDDLE  ALT=\"p=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%2Bp=1-2=-1 ALIGN=MIDDLE  ALT=\"k%2Bp=1-2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the focus of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(h,k+p\right)\"> is the point (-2,-1)\r<BR>\n" );
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document.write( "Directrix:\r<BR>\n" );
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document.write( "The directrix of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-k=a%28x-h%29%5E2 ALIGN=MIDDLE  ALT=\"y-k=a%28x-h%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k-p ALIGN=MIDDLE  ALT=\"y=k-p\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k-p=1-%28-2%29=3 ALIGN=MIDDLE  ALT=\"y=k-p=1-%28-2%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the directrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3 ALIGN=MIDDLE  ALT=\"y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Axis of Symmetry:\r<BR>\n" );
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document.write( "In this case, the axis of symmetry is simply the equation of the vertical line through the vertex.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-2 ALIGN=MIDDLE  ALT=\"h=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the axis of symmetry is <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;\" >\r<BR>\n" );
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document.write( "Visual Check:\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%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-1%2F8%29%28x%2B2%29%5E2%2B1%29%2C%0D%0Agreen%28line%28-20%2C3%2C20%2C3%29%29%2C%0D%0Ablue%28line%28-2%2C-20%2C-2%2C20%29%29%2C%0D%0Acircle%28-2%2C1%2C0.08%29%2Ccircle%28-2%2C1%2C0.10%29%2Ccircle%28-2%2C1%2C0.12%29%2C%0D%0Acircle%28-2%2C-1%2C0.08%29%2Ccircle%28-2%2C-1%2C0.10%29%2Ccircle%28-2%2C-1%2C0.12%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B8y%2B4x-4=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B8y%2B4x-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the vertex, focus, directrix (green) and axis of symmetry (blue)<BR>\n" );
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