Find an equation of the parabola 𝒚=𝒂𝒙𝟐 +𝒃𝒙+𝒄 that passes through the points (−2,4), (2,2) 𝑎𝑛𝑑 (4,9).

Answer 1

x = a(y-k)^2 + h is a horizontal parabola

plug in each point to get 3 equations 3 unknowns,

solve by elimination, substitution or matrix algebra

or try a vertical parabola, which is what the problem wants, with

y = a(x-h)^2 +k

and plug in each point

then when you find a,h, and k

rewrite in the form

y=ax^2 +bx + c

y=ax^2 -2ahx + ah^2 +k

a will be the same in both forms

b = -2ah

c = ah^2 +k

4=a(-2-h)^2 + k

2 =a(2-h)^2 + k

9= a(4-h)^2 +k

expand the above

4=4a +4ah +ah^2 +k

2=4a - 4ah +ah^2 + k

9=16a -8ah+ah^2 + k

multiply, add subtract to eliminate variables