-2x^2+8x -6:
factorise:
-2(x^2-4x+3)
(-4)/2 = -2
x^2-4x + 4 = (x-2)^2
x^2 - 4x + 3 = (x-2)^2 - 1
-2(x^2-4x+3) = -2((x-2)^2-1)
=-2(x-2)^2 + 2
a(x-h)^2 + k = -2(x-2)^2 + 2
turning point: (-h,k), where x=h is the axis of symmetry.
(-h, k) = (2,2)
x= 2 is the axis of symmetry.
since the coefficient of x^2 is negative (-2), the graph opens to the bottom.
the point (-h, k) is therefore a maximum point.
since the maximum point is the highest possible, the range is equal to or below 2.
{y: y<=2}
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6x^2+3x-18:
factorised:
6(x^2+0.5x-3)
x^2+0.5x+0.0625 = (x+0.25)^2
x^2+0.5x-3 = (x+0.25)^2-3.0625
6((x+0.25^2)-3.0625) = 6(x+0.25^2) - 18.375
a(x-h)^2 + k = 6(x+0.25)^2 - 18.375
turning point: (-h,k), where x=-h is the axis of symmetry.
(-h, k) = (-0.25,-18.375)
x= -0.25 is the axis of symmetry.
since the coefficient of x^2 is positive (6), the graph opens to the top.
(-h, k), therefore, is a minimum point.
(-h, k) = (-0.25, -18.375)
since the minimum point is the lowest possible, the range is equal to or above -18.375.
{y:y>=-18.375}
desmos.com/calculator