Question 739832
The general form is:
{{{ y = ax^2 + b*x + c }}}
The vertex is at ( -2,-8 )
The formula for the x-intercept of the vertex is:
{{{ x[v] = -b/(2a) }}}
{{{ -2 = -b/(2a) }}}
{{{ -b = -4a }}}
{{{ b = 4a }}}
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So far I have
{{{ y = ax^2 + 4a*x + c }}}
Plug in the vertex point ( -2,-8 )
{{{ -8 = a*(-2)^2 + 4a*(-2) + c }}}
{{{ -8 = 4a - 8a + c }}}
{{{ -8 = -4a + c }}}
{{{ c = 4a - 8 }}}
Now I have
{{{ y = ax^2 + 4a*x + 4a - 8 }}}
Plug in ( 0,4 )
{{{ 4 = a*0^2 + 4a*0 + 4a - 8 }}}
{{{ 4 = 4a - 8 }}}
{{{ 4a = 12 }}}
{{{ a = 3 }}}
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{{{ y = ax^2 + 4a*x + 4a - 8 }}}
{{{ y = 3x^2 + 4*3*x + 4*3 - 8 }}}
{{{ y = 3x^2 + 12x + 4 }}}
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Use the quadratic formula
{{{ x = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 3 }}}
{{{ b = 12 }}}
{{{ c = 4 }}}
{{{ x = (-12 +- sqrt( 12^2 - 4*3*4 )) / (2*3) }}}
{{{ x = (-12 +- sqrt( 144 - 48 )) / 6 }}}
{{{ x = (-12 +- sqrt( 96 )) / 6 }}}
{{{ x = ( -12 + 4*sqrt(6) ) / 6 }}}
{{{ x = ( -6 + 2*sqrt(6) ) / 3 }}}
{{{ x = ( -6 + 2*2.4495 ) / 3 }}}
{{{ x = ( -6 + 4.89898 ) / 3 }}}
{{{ x = -1.10102 / 3 }}}
{{{ x = -.367007 }}}
and
{{{ x = ( -6 - 4.89898 ) / 3 }}}
{{{ x = -10.89898 / 3 }}}
{{{ x = -3.63299 }}}
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The x-intercepts are at -.367007 and -3.63299
check:
Is the y-intercept at ( 0,4 ) ?
{{{ 4 = 3*0^2 + 12*0 + 4 }}}
{{{ 4 = 4 }}}
OK
Does it go through ( -2,-8 ) ?
{{{ y = 3x^2 + 12x + 4 }}}
{{{ -8 = 3(-2)^2 + 12*(-2) + 4 }}}
{{{ -8 = 3*4 - 24 + 4 }}}
{{{ -8 = -12 + 4 }}}
{{{ -8 = -8 }}}
OK
Here's the plot:
{{{ graph( 400, 400, -6, 3, -10, 10, 3x^2 + 12x + 4 ) }}}