Let's analyze the given function and its translation step-by-step.
We're given the function [tex]\( f(x) = -|x + 9| - 1 \)[/tex]. This is an absolute value function with a vertical shift and reflection.
1. Original Function Analysis
- The term [tex]\( |x + 9| \)[/tex] represents a horizontal shift of the standard absolute value function [tex]\( |x| \)[/tex] to the left by 9 units.
- The negative sign in front of [tex]\( |x + 9| \)[/tex] reflects the graph of the function across the x-axis.
- Subtracting 1 (i.e., [tex]\(-1\)[/tex]) shifts the entire graph downward by 1 unit.
Next, we need to translate this function 6 units up.
2. Translation Upward
- Translating a function [tex]\( g(x) \)[/tex] upward by [tex]\( k \)[/tex] units results in [tex]\( g(x) + k \)[/tex].
- Therefore, if we translate [tex]\( f(x) \)[/tex] upward by 6 units, we get [tex]\[ f(x) + 6 = -|x + 9| - 1 + 6 = -|x + 9| + 5. \][/tex]
So, the translated function becomes [tex]\( f_{trans}(x) = -|x + 9| + 5 \)[/tex].
3. Example Calculation
Let's consider an [tex]\( x \)[/tex]-value to see the effect of this translation. We'll choose [tex]\( x = 0 \)[/tex] (an arbitrary choice to illustrate the process):
- For the original function [tex]\( f(x) = -|x + 9| - 1 \)[/tex]:
[tex]\[
f(0) = -|0 + 9| - 1 = -9 - 1 = -10.
\][/tex]
- For the translated function [tex]\( f_{trans}(x) = -|x + 9| + 5 \)[/tex]:
[tex]\[
f_{trans}(0) = -|0 + 9| + 5 = -9 + 5 = -4.
\][/tex]
4. Results
After calculating, the value of the original function [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex] is [tex]\(-10\)[/tex], and the value of the translated function [tex]\( f_{trans}(x) \)[/tex] at [tex]\( x = 0 \)[/tex] is [tex]\(-4\)[/tex].
In conclusion, the original and translated function values at [tex]\( x = 0 \)[/tex] are [tex]\(-10\)[/tex] and [tex]\(-4\)[/tex] respectively.
