document.write( "Question 54437: Consider the parabola through the points (1,5), (2,11), and (3,-3).
\n" ); document.write( " i) Find the equation for the parabola y = ax² + bx + c.
\n" ); document.write( " ii) Find the vertex.
\n" ); document.write( "iii) find the x-intercepts.
\n" ); document.write( " iv) Find the y-intercept. \n" ); document.write( "
Algebra.Com's Answer #36692 by Edwin McCravy(20086) $\"\"$ $\"About$
You can put this solution on YOUR website!
Consider the parabola through the points (1,5), (2,11), and (3,-3).\r\n" );
document.write( "  i) Find the equation for the parabola y = ax² + bx + c.\r\n" );
document.write( " ii) Find the vertex.\r\n" );
document.write( "iii) find the x-intercepts.\r\n" );
document.write( " iv) Find the y-intercept.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "i)  Substitute (x,y) = (1,5) in y = ax² + bx + c\r\n" );
document.write( "\r\n" );
document.write( "         5 = a(1)² + b(1) + c\r\n" );
document.write( "         5 = a + b + c\r\n" );
document.write( "OR \r\n" );
document.write( "a + b + c = 5 \r\n" );
document.write( "\r\n" );
document.write( "Substitute (x,y) = (2,11) in y = ax² + bx + c\r\n" );
document.write( "\r\n" );
document.write( "        11 = a(2)² + b(2) + c\r\n" );
document.write( "        11 = 4a + 2b + c\r\n" );
document.write( "OR \r\n" );
document.write( "4a + 2b + c = 11\r\n" );
document.write( "\r\n" );
document.write( "Substitute (x,y) = (3,-3) in y = ax² + bx + c\r\n" );
document.write( "\r\n" );
document.write( "        -3 = a(3)² + b(3) + c\r\n" );
document.write( "        -3 = 9a + 3b + c\r\n" );
document.write( "OR \r\n" );
document.write( "9a + 3b + c = -3\r\n" );
document.write( "\r\n" );
document.write( "So now you have a system of three equations in\r\n" );
document.write( "three unknowns a, b, and c.\r\n" );
document.write( "\r\n" );
document.write( " a +  b + c =  5\r\n" );
document.write( "4a + 2b + c = 11\r\n" );
document.write( "9a + 3b + c = -3\r\n" );
document.write( "\r\n" );
document.write( "Can you solve that system?  If not, post again.\r\n" );
document.write( "\r\n" );
document.write( "Solution to system:  a = -10, b = 36, c = -21\r\n" );
document.write( "\r\n" );
document.write( "Substituting in \r\n" );
document.write( "\r\n" );
document.write( "y = ax² + bx + c\r\n" );
document.write( "\r\n" );
document.write( "y = -10x² + 36x - 21\r\n" );
document.write( "\r\n" );
document.write( "---------------------------------\r\n" );
document.write( "\r\n" );
document.write( "ii)  The x-coordinate h of the vertex is found by the formula:\r\n" );
document.write( "\r\n" );
document.write( "       h = -b/(2a)\r\n" );
document.write( "       h = -(36)/(2·-10)\r\n" );
document.write( "       h = -36/(-20)\r\n" );
document.write( "       h = 9/5\r\n" );
document.write( "\r\n" );
document.write( "iii) The y-coordinate k of the vertex is found by substituting\r\n" );
document.write( "     the x-coordinate in the equation for x and solving for y\r\n" );
document.write( "\r\n" );
document.write( "y = -10x² + 36x - 21\r\n" );
document.write( "y = -10(9/5)² + 36(9/5) - 21\r\n" );
document.write( "y = -10(81/25) + 324/5 - 21\r\n" );
document.write( "y = -162/5 + 324/5 - 105/5\r\n" );
document.write( "y = 57/5\r\n" );
document.write( "\r\n" );
document.write( "So h = 57/5\r\n" );
document.write( "\r\n" );
document.write( "The vertex is therefore (9/5, 57/5) or in decimals (1.8, 11.4)\r\n" );
document.write( "\r\n" );
document.write( "------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "iii) To find the x-intercepts, substitute 0 for y and solve for x\r\n" );
document.write( "\r\n" );
document.write( "y = -10x² + 36x - 21\r\n" );
document.write( "0 = -10x² + 36x - 21\r\n" );
document.write( "    -10x² + 36x - 21 = 0\r\n" );
document.write( "\r\n" );
document.write( "That does not factor, so we have to use the quadratic formula:\r\n" );
document.write( "                 ________ \r\n" );
document.write( "           -b ± Öb² - 4ac\r\n" );
document.write( "      x = -----------------\r\n" );
document.write( "                 2a\r\n" );
document.write( "\r\n" );
document.write( "                    ___________________ \r\n" );
document.write( "           -(36) ± Ö(36)² - 4(-10)(-21)\r\n" );
document.write( "      x = -------------------------------\r\n" );
document.write( "                     2(-10)\r\n" );
document.write( "\r\n" );
document.write( "                  __________ \r\n" );
document.write( "           -36 ± Ö1296 - 840\r\n" );
document.write( "      x = -------------------\r\n" );
document.write( "                  -20\r\n" );
document.write( "\r\n" );
document.write( "                  ___ \r\n" );
document.write( "           -36 ± Ö456\r\n" );
document.write( "      x = ------------\r\n" );
document.write( "              -20\r\n" );
document.write( "\r\n" );
document.write( "                  _____ \r\n" );
document.write( "           -36 ± Ö4·114\r\n" );
document.write( "      x = --------------\r\n" );
document.write( "               -20\r\n" );
document.write( "\r\n" );
document.write( "                   ___ \r\n" );
document.write( "           -36 ± 2Ö114\r\n" );
document.write( "      x = --------------\r\n" );
document.write( "               -20\r\n" );
document.write( "     \r\n" );
document.write( "Make two fractions:\r\n" );
document.write( "                     ___\r\n" );
document.write( "           -36     2Ö114\r\n" );
document.write( "      x = ----- ± ---------\r\n" );
document.write( "           -20      -20\r\n" );
document.write( "\r\n" );
document.write( "                  ___\r\n" );
document.write( "           9     Ö114\r\n" );
document.write( "      x = --- ± ---------\r\n" );
document.write( "           5      10\r\n" );
document.write( "\r\n" );
document.write( "These come out to be approximately\r\n" );
document.write( "\r\n" );
document.write( "x = .7322921748 and x = 2.867707825\r\n" );
document.write( "\r\n" );
document.write( "----------------------------      \r\n" );
document.write( "  \r\n" );
document.write( "iii) To find the y-intercepts, substitute 0 \r\n" );
document.write( "     for x and solve for y\r\n" );
document.write( "\r\n" );
document.write( "y = -10x² + 36x - 21\r\n" );
document.write( "y = -10(0)² + 36(0) - 21\r\n" );
document.write( "y = 0 + 0 - 21\r\n" );
document.write( "y = -21\r\n" );
document.write( "\r\n" );
document.write( "So the y-intercept is -21.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );