Find g(x), where g(x) is the translation 4 units up of f(x)=|x|.

Question

Write your answer in the form a|x–h|+k, where a, h, and k are integers.

Answer 1

To find the function \( g(x) \), which is the translation of \( f(x) = |x| \) 4 units up, we can follow these steps:

1. The original function is \( f(x) = |x| \).
2. To translate a function upwards, we add a constant to it. In this case, since we want to translate it 4 units up, we add 4 to \( f(x) \).

So we have:

\[ g(x) = f(x) + 4 = |x| + 4 \]

3. Now we need to express \( g(x) \) in the form \( a|x - h| + k \).

From \( g(x) = |x| + 4\), we can see that:

• \( a = 1 \) (the coefficient of the absolute value)
• \( h = 0 \) (there's no horizontal shift)
• \( k = 4 \) (the vertical shift)

Thus, we can write:

\[ g(x) = 1|x - 0| + 4 \]

Therefore, in the required form \( a|x - h| + k \), we have:

\[ g(x) = 1|x - 0| + 4 \]

In summary, the values are:

• \( a = 1 \)
• \( h = 0 \)
• \( k = 4 \)

Hence, the final answer is:

\[ g(x) = |x| + 4 \]