Find g(x), where g(x) is the translation 8 units up of f(x)=x2.

Question

Write your answer in the form a(x–h)2+k, where a, h, and k are integers.
g(x)=

Answer 1

To find \( g(x) \), which represents a translation of the function \( f(x) = x^2 \) up by 8 units, we can express this mathematically.

1. Starting Function: \( f(x) = x^2 \)

2. Translation Up by 8 Units: To translate the function up by 8 units, we simply add 8 to the whole function:

   \[ g(x) = f(x) + 8 = x^2 + 8 \]

3. Rewriting in the Form a(x–h)²+k: The next step is to express \( g(x) = x^2 + 8 \) in the requested vertex form \( a(x - h)^2 + k \).

   In this case:

   • The coefficient \( a \) is 1 (since it is just \( x^2 \)).
   • The value of \( h \) is 0 (there is no horizontal shift).
   • The value of \( k \) is 8 (the upward translation).

   Therefore, we can express \( g(x) \) as:

   \[ g(x) = 1(x - 0)^2 + 8 \]

Thus, the answer is:

\[ g(x) = 1(x - 0)^2 + 8 \]

Or simply:

\[ g(x) = (x - 0)^2 + 8 \]

In the required form:

\[ g(x) = 1(x - 0)^2 + 8 \]

Where \( a = 1 \), \( h = 0 \), and \( k = 8 \).