document.write( "Question 242816: I am suppose to graph this conic section how do I do this equation (x+2)=4(y-1)^2 \n" ); document.write( "
Algebra.Com's Answer #177864 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
I am suppose to graph this conic section how do I do this equation \r
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document.write( "You must learn all the various forms for \r\n" );
document.write( "all the various conic sections, and all their\r\n" );
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document.write( "This one can be placed in standard form:\r\n" );
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document.write( "\"%28y-k%29%5E2=4p%28x-h%29\"\r\n" );
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document.write( "which is a parabola with a horizontal axis of\r\n" );
document.write( "symmetry with vertex = (h,k),\r\n" );
document.write( "distance from vertex to focus and directrix = |p|\r\n" );
document.write( "which opens right if p is positive and left if p is negative.\r\n" );
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document.write( "Now we put your equation in standard form:\r\n" );
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document.write( "\"%28x%2B2%29=4%28y-1%29%5E2\"\r\n" );
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document.write( "Swap sides:\r\n" );
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document.write( "\"4%28y-1%29%5E2=x%2B2\"\r\n" );
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document.write( "Multiply both sides by \"1%2F4\"\r\n" );
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document.write( "\"%28y-1%29%5E2=1%2F4\"\"%28x%2B2%29\"\r\n" );
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document.write( "Compare to the general standard form:\r\n" );
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document.write( "\"%28y-k%29%5E2=4p%28x-h%29\"\r\n" );
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document.write( "and we have \"k=1\", \"4p=1%2F4\", \"h=-2\"\r\n" );
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document.write( "So the vertex is (h,k) = (-2,1) and \"p=1%2F16\"\r\n" );
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document.write( "and since p is positive the parabola opens to\r\n" );
document.write( "the right.\r\n" );
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document.write( "So we plot the vertex (-2,1)\r\n" );
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document.write( "and the axis of symmetry is the horizontal line through the vertex,\r\n" );
document.write( "whose equation is \r\n" );
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document.write( "\"y=1\"\r\n" );
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document.write( "We'll draw it in blue:\r\n" );
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document.write( "Since \"p=1%2F16\", and since it is positive, the focus\r\n" );
document.write( "is \"1%2F16\" of a unit to the right of the vertex, or the point\r\n" );
document.write( "(\"-2%2B1%2F16\",1) = (\"-31%2F16\",1) = (\"-1%2615%2F16\",1)\r\n" );
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document.write( "So we plot the focus, but unfortunately, it's so close \r\n" );
document.write( "to the vertex, it's hard to plot it so it doesn't run \r\n" );
document.write( "into the vertex.\r\n" );
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document.write( "Now we draw the directrix which is a vertical line \"1%2F16\" of a unit to the left of the vertex.  Its equation is \r\n" );
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document.write( "\"x+=+-33%2F16\" \r\n" );
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document.write( "We'll draw it in green:\r\n" );
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document.write( "Next we draw a horizontal line from the focus directly through\r\n" );
document.write( "the vertex to the directrix. Unfortunately you can hardly see it \r\n" );
document.write( "because it's so short because those points are so close together:\r\n" );
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document.write( "  \r\n" );
document.write( "Draw a square on each side of that line.  They too are unfortunately\r\n" );
document.write( "very tiny in this problem:\r\n" );
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document.write( "Now we can find the y-intercepts by letting x = 0 in the\r\n" );
document.write( "original equation and solving for y\r\n" );
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document.write( "\"%28x%2B2%29=4%28y-1%29%5E2\"\r\n" );
document.write( "\"%280%2B2%29=4%28y-1%29%5E2\"\r\n" );
document.write( "\"2=4%28y-1%29%5E2\"\r\n" );
document.write( "\"1%2F2=%28y-1%29%5E2\"\r\n" );
document.write( "±\"sqrt%281%2F2%29=y-1\"\r\n" );
document.write( "\"1%2B-sqrt%281%2F2%29=y\"\r\n" );
document.write( "This gives the y-intercepts at about (0,1.7) and (0.3)\r\n" );
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document.write( "So we plot those:\r\n" );
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document.write( "Finally we draw the parabola through the right corners of the\r\n" );
document.write( "squares, through the vertex and through the y-intercepts:\r\n" );
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document.write( "That's it.  It's too bad the parabola you picked to plot had \r\n" );
document.write( "such a tiny value of p.\r\n" );
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document.write( "Edwin
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