Sure! Let's go through the process of translating the function [tex]\( f(x) = -|x + 9| - 1 \)[/tex] 6 units up, step by step.
### Step 1: Understanding the original function
The original function given is:
[tex]\[ f(x) = -|x + 9| - 1 \][/tex]
### Step 2: Translation upwards
When translating a function [tex]\( k \)[/tex] units upwards, we simply add [tex]\( k \)[/tex] to the function. In this problem, we are translating the function 6 units upwards. Thus, we need to add 6 to the original function.
### Step 3: Adding the constant to the function
We add 6 to the given function:
[tex]\[ f(x) = -|x + 9| - 1 + 6 \][/tex]
### Step 4: Simplifying the expression
Combine the constants [tex]\(-1\)[/tex] and [tex]\(+6\)[/tex]:
[tex]\[ -1 + 6 = 5 \][/tex]
So the new function becomes:
[tex]\[ f(x) = -|x + 9| + 5 \][/tex]
### Conclusion
After translating the original function 6 units up, our new function is:
[tex]\[ f(x) = -|x + 9| + 5 \][/tex]
This is the function obtained after performing the required translation.
