Question

Arrange the parabolas represented by the equations in increasing order with respect to the y-values of their directrixes. Tiles y = -x2 + 3x + 8 y = 2x2 + 15x + 18 y = x2 + 13x + 5 y = -2x2 + 4x + 8 Sequence

Answer 1

A standard form for the equation of a parabola is
y = a*(x-h)^2 + k
when the vertex is at (h,k)

Test the given parabolas by writing them in the standard form.
y = -x^2 + 3x + 8
= -(x^2 - 3x) + 8
= -[(x - 3/2)^2 - 9/4] + 8
= -(x - 3/2)^2 + 41/4
The directrix has y-value of 2*(41/4) = 82/4 = 20.5

y = 2*x^2 + 15x + 18
= 2[x^2 + (15/2)x ] + 18
= 2[(x+15/4)^2 - (15/4)^2] + 18
= 2(x + 15/4)^2 - 81/8
The directrix has y-value of 2*(-81/8) = 2*(-81/8) = -20.25

y = x^2 + 13x + 5
= (x + 13/2)^2 - (13/2)^2 + 5
= (x + 13/2)^2 - 149/4
The directrix has y-value of 2*(-149/4) = -74.5

y = -2x^2 + 4x + 8
= -2[x^2 + 2x] + 8
= -2[(x+1)^2 - 1] + 8
= -2(x+1)^2 + 10
The directrix has y-value of 2*10 = 20

In increasing order with respect to y-values of their directrixes, the equations are
y = x^2 + 13x +5
y = 2*x^2 + 15x + 18
y = -2*x^2 + 4x + 8
y = -x^2 + 3x + 8