To find the translation of $f(x) = |x|$ by 9 units left and 9 units up, we'll follow these steps:

1. Translation 9 units left: To translate the function $f(x) = |x|$ 9 units to the left, we replace $x$ with $x + 9$ inside the absolute value function. This gives: $f(x + 9) = |x + 9|$

2. Translation 9 units up: To translate the function 9 units up, we add 9 to the entire function. This results in: $g(x) = |x + 9| + 9$

Now, the function is in the form $a|x - h| + k$.

• In this case, $a = 1$, $h = -9$ (because it's $x + 9$, so $h = -9$), and $k = 9$.

Thus, the translated function is: $g(x) = |x + 9| + 9$ or equivalently: $g(x) = |x - (-9)| + 9$

Summary:

The translated function is: $g(x) = |x + 9| + 9$

Would you like further details or have additional questions?

Related questions:

1. How would you translate $f(x) = |x|$ 5 units right and 3 units down?
2. How does the translation affect the graph of the absolute value function?
3. What is the general form of a translation of a function $f(x)$ in terms of $a$, $h$, and $k$?
4. How do vertical translations and horizontal translations differ in terms of their effects on the graph?
5. What happens to the function if you translate it 9 units to the left but 5 units down instead?

Tip: Remember that horizontal translations involve modifying the input $x$ (i.e., replacing $x$ with $x + h$), while vertical translations involve adding or subtracting a constant outside the function.