document.write( "Question 202316: find (a)the directrix, (b)the focus, and (c)the roots of the parabola
\n" ); document.write( "y = x^2 - 5x + 4 \n" ); document.write( "

Algebra.Com's Answer #152693 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!
find (a)the directrix, (b)the focus, and (c)the roots of the parabola
\n" ); document.write( " $\"y+=+x%5E2+-+5x+%2B+4\"$
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "We have to get it in the form \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Swap sides:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Add -4 to both sides the get the x-terms alone on the left.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Multiply the coefficient of x, which is  by \r\n" );
document.write( "This gives .  Now we square  and get \r\n" );
document.write( "We add  to both sides:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Factor the left side, and write the  as \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Write the left side as the square of a binomial,\r\n" );
document.write( "combine the fractions on the right:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now so that equation will look like this:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "we put the right side in parentheses and put a 1\r\n" );
document.write( "coefficient before the parentheses, like this:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we can compare it to the equation:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and get , so \r\n" );
document.write( "\r\n" );
document.write( ", so \r\n" );
document.write( "\r\n" );
document.write( "So the vertex is (,) or (,)\r\n" );
document.write( "\r\n" );
document.write( "And , so \r\n" );
document.write( "\r\n" );
document.write( "Now let's begin by plotting the vertex, which is (,),\r\n" );
document.write( "\r\n" );
document.write( "But for plotting purposes, mixed numbers are better\r\n" );
document.write( "than improper fractions, so for plotting vertex (,),\r\n" );
document.write( "we rewrite it as (,)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we will find the x-intercepts, by settng \r\n" );
document.write( "in the original equation, and finding the \"roots\":\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( ", so \r\n" );
document.write( ", so \r\n" );
document.write( "\r\n" );
document.write( "So the x-intecepts are (1,0) and (4,0) \r\n" );
document.write( "\r\n" );
document.write( "So we plot those:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and sketch in the parabola:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now the focus is p units from the vertex INSIDE\r\n" );
document.write( "the parabola, so since the parabola opens upward,\r\n" );
document.write( "we add P or  to the y-coordinate of the\r\n" );
document.write( "vertex. Since the vertx is (,),\r\n" );
document.write( "the focus = (,) = (,) \r\n" );
document.write( "= (,)\r\n" );
document.write( "\r\n" );
document.write( "So we draw that point:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now the directrix is a line OUTSIDE the parabola which is\r\n" );
document.write( "also p-units, or  from the vertex.\r\n" );
document.write( "\r\n" );
document.write( "Since the y-coordinate of the vertex is  we want\r\n" );
document.write( "the directrix to be  unit below the vertex, so we\r\n" );
document.write( "subtract \r\n" );
document.write( "\r\n" );
document.write( "So the directrix is the horizontal line whose equation is \r\n" );
document.write( "\r\n" );
document.write( ".  I'll draw it in in green:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So the focus is the POINT (,) and the \r\n" );
document.write( "\r\n" );
document.write( "directrix is the LINE \r\n" );
document.write( "\r\n" );
document.write( "The \"roots\" are really the y-coordinates of the \r\n" );
document.write( "x-intercepts or 1 and 4.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );