document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204031>Question 1204031</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose a parabola opens upward, has focus at (1,-1), has a horizontal directrix, and passes through the point (-31,59).\r<BR>\n" );
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document.write( "a. The directrix will have an equation of?\r<BR>\n" );
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document.write( "b. The equation of the parabola will be?<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 #840034 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=840034\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "F = focus = (1,-1)<BR>\n" );
document.write( "P = point on parabola = (-31,59)<BR>\n" );
document.write( "D = point on directrix that is directly above or below P (this is for a horizontal directrix only)\r<BR>\n" );
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document.write( "For a parabola to be possible, we must have this condition hold:<BR>\n" );
document.write( "FP = PD\r<BR>\n" );
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document.write( "Check out the diagram shown here<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=300 src = \"https://upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Parts_of_Parabola.svg/1024px-Parts_of_Parabola.svg.png\"><BR>\n" );
document.write( "Image Source:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Parabola\">https://en.wikipedia.org/wiki/Parabola</a><BR>\n" );
document.write( "The pink congruent segments are what you should focus on.\r<BR>\n" );
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document.write( "Use the distance formula to see how far it is from the focus F(1,-1) to the point on the parabola P(-31,59)<BR>\n" );
document.write( "(x1,y1) = (1,-1) and (x2,y2) = (-31,59)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%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=d+=+sqrt%28+%281-%28-31%29%29%5E2+%2B+%28-1-59%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%281-%28-31%29%29%5E2+%2B+%28-1-59%29%5E2+%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=d+=+sqrt%28+%281%2B31%29%5E2+%2B+%28-1-59%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%281%2B31%29%5E2+%2B+%28-1-59%29%5E2+%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=d+=+sqrt%28+%2832%29%5E2+%2B+%28-60%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%2832%29%5E2+%2B+%28-60%29%5E2+%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=d+=+sqrt%28+1024+%2B+3600+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+1024+%2B+3600+%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=d+=+sqrt%28+4624+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+4624+%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=d+=+68 ALIGN=MIDDLE  ALT=\"d+=+68\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance from the focus F(1,-1) to a point on the parabola P(-31,59) is exactly 68 units.<BR>\n" );
document.write( "This is the length of segment FP.<BR>\n" );
document.write( "It's also the length of segment PD.\r<BR>\n" );
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document.write( "Due to the horizontal directrix, from point P(-31,59), we will either move up 68 units or move down 68 units to arrive at point D.<BR>\n" );
document.write( "Because P has a y coordinate larger than the y coordinate of F, it must mean the parabola opens upward. Consequently it means the directrix is below point P.<BR>\n" );
document.write( "So we'll move down 68 units from P(-31,59) to D(-31,-9)<BR>\n" );
document.write( "The equation of the directrix is then the y coordinate of point D.\r<BR>\n" );
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document.write( "<font color=red>Answer</font>: The equation of the directrix is y = -9\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Start at the focus F(1,-1) and move straight down to arrive at the directrix y = -9<BR>\n" );
document.write( "This will move you 8 units down.<BR>\n" );
document.write( "Half of which is 4.<BR>\n" );
document.write( "This is the focal length. It's the distance between vertex and focus. It's also the distance from vertex to directrix.\r<BR>\n" );
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document.write( "Start at F(1,-1) to move 4 units down to arrive at the vertex V(1,-5)\r<BR>\n" );
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document.write( "Recall the vertex form template is<BR>\n" );
document.write( "y = a(x-h)^2+k<BR>\n" );
document.write( "where<BR>\n" );
document.write( "a = determines whether the parabola opens up or down<BR>\n" );
document.write( "(h,k) = vertex\r<BR>\n" );
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document.write( "Plug in the coordinates of the vertex to get<BR>\n" );
document.write( "y = a(x-h)^2+k<BR>\n" );
document.write( "y = a(x-1)^2+(-5)<BR>\n" );
document.write( "y = a(x-1)^2-5\r<BR>\n" );
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document.write( "Then plug in the coordinates of the other point on the parabola (-31,59). This will allow us to solve for 'a'<BR>\n" );
document.write( "y = a(x-1)^2-5<BR>\n" );
document.write( "59 = a(-31-1)^2-5<BR>\n" );
document.write( "59 = a(-32)^2-5<BR>\n" );
document.write( "59 = 1024a-5<BR>\n" );
document.write( "1024a-5 = 59<BR>\n" );
document.write( "1024a = 59+5<BR>\n" );
document.write( "1024a = 64<BR>\n" );
document.write( "a = 64/1024<BR>\n" );
document.write( "a = (1*64)/(16*64)<BR>\n" );
document.write( "a = 1/16<BR>\n" );
document.write( "a = 0.0625 exactly\r<BR>\n" );
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document.write( "Therefore the vertex form equation would be<BR>\n" );
document.write( "y = (1/16)(x-1)^2-5<BR>\n" );
document.write( "or<BR>\n" );
document.write( "y = 0.0625(x-1)^2-5\r<BR>\n" );
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document.write( "I'll expand out the decimal form<BR>\n" );
document.write( "y = 0.0625(x-1)^2-5<BR>\n" );
document.write( "y = 0.0625(x^2-2x+1)-5<BR>\n" );
document.write( "y = 0.0625x^2-0.125x+0.0625-5<BR>\n" );
document.write( "y = 0.0625x^2-0.125x-4.9375<BR>\n" );
document.write( "This is in standard form y = ax^2+bx+c<BR>\n" );
document.write( "a = 0.0625 = 1/16<BR>\n" );
document.write( "b = -0.125 = -1/8<BR>\n" );
document.write( "c = -4.9375 = -79/16<BR>\n" );
document.write( "Each decimal is exact and hasn't been rounded.\r<BR>\n" );
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document.write( "<font color=red>Answer</font><BR>\n" );
document.write( "The equation of the parabola in standard form, using fractions, is y = (1/16)x^2 - (1/8)x - 79/16<BR>\n" );
document.write( "That converts to the exact decimal form y = 0.0625x^2-0.125x-4.9375\r<BR>\n" );
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document.write( "Or to save time, you can stick to the vertex form y = (1/16)(x-1)^2-5 aka y = 0.0625(x-1)^2-5<BR>\n" );
document.write( "It will depend on which form your teacher will want.<BR>\n" );
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