To find the translation of $f(x) = |x|$ that moves 5 units to the left and 5 units up, we need to apply transformations accordingly:

1. Left Translation by 5 Units: Shifting left means we add 5 to $x$. So, we replace $x$ with $x + 5$ in the function.

   • This gives $|x + 5|$.

2. Up Translation by 5 Units: Shifting up means we add 5 to the entire function.

   • This gives $|x + 5| + 5$.

Thus, the translated function $g(x)$ is: $g(x) = |x + 5| + 5$

This is already in the form $a|x - h| + k$, where:

• $a = 1$,
• $h = -5$,
• $k = 5$.

Would you like more details or have any questions?

Here are five related questions:

1. How do different translations (left, right, up, down) affect the graph of an absolute value function?
2. What would happen if we reflected the graph of $f(x) = |x|$ across the x-axis?
3. How can we identify the vertex of a translated absolute value function?
4. How do we describe horizontal and vertical stretches or compressions of $|x|$?
5. What effect does changing the value of $a$ have on the graph of $a|x|$?

Tip: Translating a function left or right involves changing the input (inside the absolute value), while translating up or down affects the output (outside the absolute value).