document.write( "Question 1110500: Determine a quadratic function for y = ax^2+bx+c if its graph passes through the point (2,19) and it has a horizontal tangent at (-1,-8). \r
\n" ); document.write( "\n" ); document.write( "I would appreciate it if someone could help me with this! \n" ); document.write( "

Algebra.Com's Answer #725516 by josmiceli(19441) $\"\"$ $\"About$

You can put this solution on YOUR website!
If the tangent is horizontal, then the only
\n" ); document.write( "place it can be is at the vertex because no
\n" ); document.write( "other point on a quadratic has a horizontal
\n" ); document.write( "slope ( slope = zero )
\n" ); document.write( "----------------------------------------
\n" ); document.write( " $\"+y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+\"$
\n" ); document.write( "The horizontal tangent is at ( -1, -8 )
\n" ); document.write( "The formula for the x-value of the vertex is:
\n" ); document.write( " $\"+x%5Bv%5D+=+-b%2F%282a%29+\"$
\n" ); document.write( " $\"+-1+=+-b%2F%282a%29+\"$
\n" ); document.write( " $\"+b+=+2a+\"$
\n" ); document.write( "and
\n" ); document.write( "( -1, -8 )
\n" ); document.write( " $\"+-8+=+a%2A%28-1%29%5E2+%2B+b%2A%28-1%29+%2B+c+\"$
\n" ); document.write( " $\"+-8+=+a+-+b+%2B+c+\"$
\n" ); document.write( " $\"+-8+=+a+-2a+%2B+c+\"$
\n" ); document.write( " $\"+-a+%2B+c+=+-8+\"$
\n" ); document.write( "(1) $\"+a+-+c+=+8+\"$
\n" ); document.write( "----------------------
\n" ); document.write( "( 2, 19 )
\n" ); document.write( " $\"+y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+\"$
\n" ); document.write( " $\"+19+=+a%2A2%5E2+%2B+b%2A2+%2B+c+\"$
\n" ); document.write( " $\"+19+=+4a+%2B+2b+%2B+c+\"$
\n" ); document.write( " $\"+19+=+4a+%2B+2%2A%282a%29+%2B+c+\"$
\n" ); document.write( "(2) $\"+8a+%2B+c+=+19+\"$
\n" ); document.write( "Add equations (1) and (2)
\n" ); document.write( "(1) $\"+a+-+c+=+8+\"$
\n" ); document.write( "(2) $\"+8a+%2B+c+=+19+\"$
\n" ); document.write( "--------------------------
\n" ); document.write( " $\"+9a+=+27+\"$
\n" ); document.write( " $\"+a+=+3+\"$
\n" ); document.write( "and
\n" ); document.write( " $\"+b+=+2a+\"$
\n" ); document.write( " $\"+b+=+2%2A3+\"$
\n" ); document.write( " $\"+b+=+6+\"$
\n" ); document.write( "and
\n" ); document.write( "(2) $\"+8a+%2B+c+=+19+\"$
\n" ); document.write( "(2) $\"+8%2A3+%2B+c+=+19+\"$
\n" ); document.write( "(2) $\"+24+%2B+c+=+19+\"$
\n" ); document.write( "(2) $\"+c+=+-5+\"$
\n" ); document.write( "------------------------------
\n" ); document.write( "The equation is:
\n" ); document.write( " $\"+y+=+3x%5E2+%2B+6x+-+5+\"$
\n" ); document.write( "------------------------------
\n" ); document.write( "check answer:
\n" ); document.write( "( 2, 19 )
\n" ); document.write( " $\"+19+=+3%2A2%5E2+%2B+6%2A2+-+5+\"$
\n" ); document.write( " $\"+19+=+3%2A4+%2B+12+-+5+\"$
\n" ); document.write( " $\"+19+=+24+-+5+\"$
\n" ); document.write( " $\"+19+=+19+\"$
\n" ); document.write( "OK
\n" ); document.write( "(-1,-8 )
\n" ); document.write( " $\"+-8+=+3%2A%28-1%29%5E2+%2B+6%2A%28-1%29+-5+\"$
\n" ); document.write( " $\"+-8+=+3+-+6+-+5+\"$
\n" ); document.write( " $\"+-8+=+-8+\"$
\n" ); document.write( "OK
\n" ); document.write( "Here's the plot:
\n" ); document.write( " $\"+graph%28+400%2C+400%2C+-6%2C+6%2C+-10%2C+10%2C+3x%5E2+%2B+6x+-+5+%29+\"$
\n" ); document.write( "Feel free to get another opinion on this\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );