
Graph n(x)=-1-|x|             the meeting point meets on the x axis at (0,-1)and decreases at (1,-2)
 on the Cartesian plane to the right.

  is increasing on the interval(s):  
Incorrect  

 
  is decreasing on the interval(s):

The given function is \( n(x) = -1 - |x| \). Let's analyze the behavior of this function.

### Function Description:
1. The function involves an absolute value, which means it has symmetry about the y-axis.
2. The term \( -|x| \) makes the function "V-shaped," but it's reflected downwards because of the negative sign.
3. The function is shifted down by 1 unit due to the \( -1 \) term.

### Key Points:
- At \( x = 0 \), \( n(x) = -1 \), which corresponds to the point \( (0, -1) \), as described in the problem.
- As \( x \) moves away from 0 (both to the left and the right), the function decreases because the absolute value term causes the function to become more negative.

### Decreasing Intervals:
- The function is **decreasing on both sides of \( x = 0 \)** because \( |x| \) is always increasing as \( |x| \) gets larger. Since the absolute value is negative, it makes \( n(x) \) more negative.
- Therefore, the function decreases for all values of \( x \) as you move away from 0. 

Thus, the function is decreasing on the interval:

\[
\text{Decreasing on the interval(s): } (-\infty, \infty)
\]

There are no increasing intervals, as the function decreases everywhere.

### Do you want details or have any questions?

Here are 5 relative questions:
1. How would you graph the function \( n(x) = -1 - |x| \)?
2. What does the symmetry about the y-axis indicate for this function?
3. How does the negative sign affect the absolute value function's shape?
4. What happens to \( n(x) \) as \( x \to \infty \) or \( x \to -\infty \)?
5. How does a vertical shift change the graph of a function?

**Tip:** When graphing absolute value functions, always check for symmetry and pay attention to shifts and reflections caused by coefficients.

Graph n(x) = -1 - |x|. The meeting point is on the x-axis at (0, -1) and decreases at (1, -2) on the Cartesian plane to the right. Determine the interval(s) where the function is increasing or decreasing.

Explore the decreasing nature of the function n(x) = -1 - |x|, a V-shaped graph reflected downward with symmetry about the y-axis. Learn key concepts like graphing absolute value functions and identifying intervals where the function decreases. Suitable for students in Grades 9-12.

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