Question

The graph shows g(x), which is a translation of f(x)=x2. 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), (-3, 1), etc.
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. Using the three points (1,3), (−3,1), (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 and second:
a(1 − 2h) = 11
a(6h + 9) = 9

From these 3/(2h+3) = a and 3(1 − 2h) = 11(2h + 3), which gives h = −15/14. Then a = 7/2 and k = −8 − a h^2 = −673/56.

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