Question 1085857
The general formula is:
{{{ y = a*x^2 + b*x + c }}}
The formula for the vertex is:
{{{ x[v] = -b/(2a) }}}
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You are given 2 points:
( -3,16 )
( 0, -20 )
Plug these values into the general formula
(-3,16)
{{{ 16 = a*(-3)^2 + b*(-3) + c }}}
{{{ 9a - 3b + c = 16 }}}
(0,-20)
{{{ -20 = a*0^2 + b*0 + c }}}
{{{ c = -20 }}}
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Now the equation is:
{{{ y = a*x^2 + b*x - 20 }}}
Al;so:
{{{ x[v] = -b/(2a) }}}
{{{ x[v] = -3 }}} ( given )
{{{ -3 = -b/(2a) }}}
{{{ -b = -6a }}}
{{{ b = 6a }}}
So, now I have:
{{{ y = a*x^2 + (6a)*x - 20 }}}
also
{{{ y[v] = 16 }}} ( given )
{{{ 16 = a*(-3)^2 + (6a)*(-3) - 20 }}}
{{{ 16 = 9a - 18a - 20 }}}
{{{ -9a = 36 }}}
{{{ a = -4 }}}
and
{{{ b = 6a }}}
{{{ b = -24 }}}
so, the equation is:
{{{ y = -4x^2 - 24x - 20 }}}
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Find the roots:
{{{x[1] = (-b + sqrt( b^2-4*a*c ))/(2*a) }}}
{{{x[1] = (-(-24) + sqrt( (-24)^2-4*a*c ))/(2*a) }}}
{{{x[1] = ( 24 + sqrt( 576 - 4*(-4)*(-20) ))/(2*(-4)) }}}
{{{x[1] = ( 24 + sqrt( 576 - 320 ))/(-8) }}}
{{{x[1] = ( 24 + sqrt( 256 ))/(-8) }}}
{{{x[1] = ( 24 + 16 )/(-8) }}}
{{{x[1] = 40/(-8) }}}
{{{x[1] = -5 }}}
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The other root is at:
{{{ x[2] = ( 24 - 16 )/(-8) }}}
{{{ x[2] = 8/(-8) }}}
{{{ x[2] = -1 }}}
The x-intercepts are:
( -5,0 ) and ( -1,0 )
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Here's the plot:
{{{ graph( 400, 400, -10, 2, -5, 20, -4x^2 - 24x - 20 ) }}}
check the math and get an other opinion if needed