To analyze the function [tex]\( f(x) = |x + 4| - 2 \)[/tex], we'll break it down into several steps: graphing it and then determining the intervals where the function is increasing, decreasing, or constant.
Step-by-Step Solution:
### 1. Understanding the Function
The function [tex]\( f(x) = |x + 4| - 2 \)[/tex] is a transformation of the absolute value function [tex]\( y = |x| \)[/tex].
- Absolute Value Basic Form: [tex]\( y = |x| \)[/tex]
- Horizontal Shift: [tex]\( y = |x + 4| \)[/tex] moves the graph 4 units to the left.
- Vertical Shift: Subtracting 2, resulting in [tex]\( y = |x + 4| - 2 \)[/tex], moves the graph 2 units downward.
### 2. Graphing the Function
Start by graphing the function [tex]\( y = |x| \)[/tex], which is V-shaped with its vertex at the origin (0, 0). Shift this graph left by 4 units to get [tex]\( y = |x + 4| \)[/tex], putting the vertex at (-4, 0). Finally, shift the graph down by 2 units to get [tex]\( f(x) = |x + 4| - 2 \)[/tex], placing the vertex at (-4, -2).
### 3. Sketch the Graph
The key points to sketch the function are:
- The vertex at (-4, -2)
- For [tex]\( x < -4 \)[/tex], [tex]\( f(x) = -(x + 4) - 2 = -x - 4 - 2 = -x - 6 \)[/tex]
- For [tex]\( x > -4 \)[/tex], [tex]\( f(x) = (x + 4) - 2 = x + 2 \)[/tex]
Now, sketch the graph:
- For [tex]\( x < -4 \)[/tex], the line is [tex]\( y = -x - 6 \)[/tex] (downward slope).
- For [tex]\( x > -4 \)[/tex], the line is [tex]\( y = x + 2 \)[/tex] (upward slope).
### 4. Determine Intervals of Increase and Decrease
From the graph, it is evident:
- Increasing Interval: After the vertex (-4, -2), the function [tex]\( f(x) \)[/tex] increases as x increases. Hence, the increasing interval is [tex]\( (-4, \infty) \)[/tex].
- Decreasing Interval: Before the vertex (-4, -2), the function [tex]\( f(x) \)[/tex] decreases as x decreases. Hence, the decreasing interval is [tex]\( (-\infty, -4) \)[/tex].
### Conclusion
Increasing: [tex]\( (-4, \infty) \)[/tex]
Decreasing: [tex]\( (-\infty, -4) \)[/tex]
There are no intervals where the function is constant, as the only point of constancy is exactly at [tex]\( x = -4 \)[/tex], where the vertex is located.
