To find the function $g(x)$ that represents the translation of $f(x) = |x|$ 8 units up, we modify the equation by adding 8 to the output of the function.

The original function is:

$f(x) = |x|$

When translating a function upward by $k$ units, we add $k$ to the function:

$g(x) = f(x) + 8 = |x| + 8$

Now, the given form is $a|x - h| + k$. In this case:

• $a = 1$ (since the slope of the absolute value is unchanged),
• $h = 0$ (there's no horizontal shift),
• $k = 8$ (the vertical translation).

Thus, the function becomes:

$g(x) = 1|x - 0| + 8$

Simplified:

$g(x) = |x| + 8$

Would you like further details or have any questions?

Here are 5 related questions for you:

1. How would the equation change if we translated $f(x) = |x|$ 5 units down instead?
2. What happens to the graph of $f(x) = |x|$ when translated 3 units to the right?
3. How do you find the horizontal translation of $f(x) = |x|$?
4. If $f(x) = 2|x|$, what is the translation 8 units up?
5. What are the effects of translating a function both horizontally and vertically?

Tip: Vertical translations simply add or subtract from the function's output, while horizontal translations affect the input inside the function.