Question 1061430
{{{ f(x) = 4x^2 + 4x + 6 }}}
You find the y-intercept by making {{{ x = 0 }}}
{{{ f(x) = f(0) }}}
{{{ f(0) = 4*0^2 + 4*0 + 6 }}}
{{{ f(0) = 6 }}}
The y-intercept is at ( 0, 6 )
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You find the x-intercepts by making {{{ f(x) = 0 }}}
{{{ 0 = 4x^2 + 4x + 6 }}}
divide both sides by {{{ 2 }}}
{{{ 0 = 2x^2 + 2x + 3 }}}
Use the quadratic formula
when the form is:
{{{ f(x) = a*x^2 + b*x + c }}}
{{{ x = ( -b +- sqrt( b^2 - 4*a*c ))/(2*a) }}}
{{{ a = 2 }}}
{{{ b = 2 }}}
{{{ c = 3 }}}
{{{ x = ( -2 +- sqrt( 2^2 - 4*2*3 ))/(2*2) }}}
{{{ x = ( -2 +- sqrt( 4 - 24 ))/4 }}}
{{{ x = ( -2 +- sqrt( -20 ) )/4 }}}
{{{ x = ( -2 + 2*sqrt(5)*i ) / 4 }}}
{{{ x = -1/2 + ( sqrt(5)/2 )*i }}}
and, also:
{{{ x = -1/2 - ( sqrt(5)/2 )*i }}}
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The roots in this case are a pair
of imaginaries called a complex conjugate pair.
There are no actual intersections with the x-axis.
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Here's the plot so you can see the y-intercept,
and the fact that there is no intersection with x-axis
{{{ graph( 400, 500, -5, 5, -2, 15, 4x^2 + 4x + 6 ) }}}