Question

The graph shows g(x), which is a translation of f(x)=x^2. Write the function rule for g(x).
The graph shows a curved line that has a y-intercept at -8, as well as distinct points of (1, 3) and (-3, 1).
Write your answer in the form a(x–h)^2+k, where a, h, and k are integers or simplified fractions.

Answer 1

Let g(x) = a(x − h)^2 + k. Use g(1)=3, g(-3)=1, g(0)=-8:

a(1−h)^2 + k = 3
a(−3−h)^2 + k = 1
a h^2 + k = −8

Subtract the third from the first two:
a(1−2h) = 11 => a = 11/(1−2h)
a(9+6h) = 9 => a = 9/(9+6h)

Solve 11/(1−2h) = 9/(9+6h) → h = −15/14. Then a = 7/2 and k = −8 − a h^2 = −673/56.

So
g(x) = (7/2)(x + 15/14)^2 − 673/56.