Translation 5 Units Up of f(x) = x^2 — find g(x) in vertex form a(x–h)^2+k

Problem

Find g(x), where g(x) is the translation 5 units up of f(x) = x^2. Write your answer in the form a(x–h)^2 + k, where a, h, and k are integers.

Step-by-step solution

1. Start with f(x) = x^2. This is a parabola with vertex at (0, 0) and can be written in vertex form as 1(x - 0)^2 + 0.
2. A translation 5 units up adds 5 to every y-value. Algebraically, add 5 to the function: g(x) = x^2 + 5.
3. Write g(x) in vertex form a(x - h)^2 + k. Using the vertex (0, 5), we have a = 1, h = 0, k = 5, so g(x) = 1(x - 0)^2 + 5.

Final answer

g(x) = 1(x - 0)^2 + 5, which simplifies to g(x) = x^2 + 5.