document.write( "Question 1035268: Find the vertex, focus, and directrix of the parabola with equation y=x^2+6x+5 \n" ); document.write( "
Algebra.Com's Answer #649941 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\"y=x%5E2%2B6x%2B5\"
\n" ); document.write( "
\r\n" );
document.write( "Since the coefficient of x2 is positive,\r\n" );
document.write( "we know that the parabola opens upward.\r\n" );
document.write( "\r\n" );
document.write( "We must get it into the form:\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-h%29%5E2=4p%28y-k%29\"\r\n" );
document.write( "\r\n" );
document.write( "where (h,k) is the vertex and p is the number of units\r\n" );
document.write( "the vertex is from both the focus (point) and the \r\n" );
document.write( "directrix (line).\r\n" );
document.write( "\r\n" );
document.write( "\"y=x%5E2%2B6x%2B5\"\r\n" );
document.write( "\r\n" );
document.write( "Get the x terms on the left and other terms on the right\r\n" );
document.write( "\r\n" );
document.write( "\"-x%5E2-6x=-y%2B5\"\r\n" );
document.write( "\r\n" );
document.write( "Multiply through by -1\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2%2B6x=y-5\"\r\n" );
document.write( "\r\n" );
document.write( "Complete the square on the left:\r\n" );
document.write( "1. Multiply the coefficient of x by 1/2, getting 6(1/2) = 3\r\n" );
document.write( "2. Square the result of step 1, getting +9\r\n" );
document.write( "3. Add +9 to both sides \r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2%2B6x%2B9=y-5%2B9\"\r\n" );
document.write( "\r\n" );
document.write( "Factor the left side, combine like terms on the right:\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%2B3%29%28x%2B3%29=y%2B4\"\r\n" );
document.write( "\r\n" );
document.write( "Write (x+3)(x+3) as (x+3)2\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%2B3%29%5E2=y%2B4\"\r\n" );
document.write( "\r\n" );
document.write( "Factor out 1 on the right to show the value of 4p\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%2B3%29%5E2=1%28y%2B4%29\"\r\n" );
document.write( "\r\n" );
document.write( "Compare to\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-h%29%5E2=4p%28y-k%29\"\r\n" );
document.write( "\r\n" );
document.write( "-h=+3 so h=-3\r\n" );
document.write( "4p=1 so p=1/4\r\n" );
document.write( "-k=+4 so k=-4\r\n" );
document.write( "\r\n" );
document.write( "Vertex = (h,k) = (-3,-4)\r\n" );
document.write( "\r\n" );
document.write( "We can easily get the intercepts from\r\n" );
document.write( "the original equation.\r\n" );
document.write( "\r\n" );
document.write( "\"y=x%5E2%2B6x%2B5\"\r\n" );
document.write( "\"x%5E2%2B6x%2B5=0\"\r\n" );
document.write( "\"%28x%2B5%29%28x%2B1%29=0\"\r\n" );
document.write( "So the x-intercepts are (-5,0) and (-1,0)\r\n" );
document.write( "\r\n" );
document.write( "To get the y-intercept:\r\n" );
document.write( "\r\n" );
document.write( "\"y=x%5E2%2B6x%2B5\"\r\n" );
document.write( "\"y=0%5E2%2B6%2A0%2B5\"\r\n" );
document.write( "\"y=5\"\r\n" );
document.write( "\r\n" );
document.write( "So the y-intercept is (0,5), so we draw\r\n" );
document.write( "the graph:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The focus is the point which is \"p=1%2F4\" of a unit above the\r\n" );
document.write( "vertex, and the directrix is a line \"p=1%2F4\" of a unit below\r\n" );
document.write( "the vertex. I'll make them green:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Since the focus (the green point) is p=1/4 of a unit above \r\n" );
document.write( "the vertex (-3,-4), the focus has the same x-coordinate as \r\n" );
document.write( "the vertex, and the y-coordinate is 1/4 of a unit above the \r\n" );
document.write( "the y-coordinate of the vertex. So we add 1/4 to the y-coordinate \r\n" );
document.write( "\"-4%2B1%2F4=-16%2F4%2B1%2F4=-15%2F4\"\r\n" );
document.write( "\r\n" );
document.write( "So the focus (the green point) is the point \"%28matrix%281%2C3%2C-3%2C%22%2C%22%2C-15%2F4%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "The directrix (the green line) is a horizontal line which is\r\n" );
document.write( "1/4 of a unit below the y-coordinate of the vertex, so we subtract\r\n" );
document.write( "1/4 from the y-coordinate of the vertex to find out how far below\r\n" );
document.write( "the x-axis the directrix is.\r\n" );
document.write( "\"-4-1%2F4=-16%2F4-1%2F4=-17%2F4\"\r\n" );
document.write( "\r\n" );
document.write( "So the directrix is the horizontal line which has the equation:\r\n" );
document.write( "\r\n" );
document.write( "\"y+=+-17%2F4\"\r\n" );
document.write( "   \r\n" );
document.write( "Edwin
\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );