Determine the equation of a quadratic in factored form that has the following characteristics: a. X-intercepts are -1 and 5 b. The optimal value is -7

Question

Need help finding equation

Answer 1

y+7 = (7/9)(x-2)^2

has vertex at (2,-7) where -7 is the minimum point

IF the problem meant for -7 as the "optimal point" to be a maximum y value then there could be no x intercepts, since x intercepts are points with y=0 > -7

y+7 = (7/9)(x-2)^2 has x intercepts at -1 and 5, the points (-1,0) and (5,0)

0+7 = (7/9)(5-2)^2 = (7/9)(9) = 7

IF the maximum point were (2,7) then the parabola with x intercepts -1 and 5 would have a vertex at (2,7) with a downward opening parabola of the form y-h = a(x-k)^2 with a<0, and vertex (k,h) = (2,7)

plug in either x intercept to calculate the value of a.

0-7 = a(5-2)^2 = 9a

a = -7/9

the equation is then

y-7 = (-7/9)(x-2)^2 with x intercepts -1 and 5 and maximum y value = 7