The question in the image asks you to graph the function $y = -|x + 2| - 2$ by plotting at least five points. Here's how to proceed:

Step 1: Understand the Function

The equation is $y = -|x + 2| - 2$:

• The term $|x + 2|$ represents an absolute value function, which creates a V-shaped graph.
• The negative sign before $|x + 2|$ reflects the graph over the x-axis, making it an upside-down V.
• The $+2$ inside the absolute value shifts the graph 2 units to the left.
• The $-2$ outside the absolute value shifts the graph 2 units downward.

Step 2: Create a Table of Points

Select values for $x$ to compute corresponding $y$ values. Here’s the process:

1. $x = -4$:
   $y = -|-4 + 2| - 2 = -| -2 | - 2 = -2 - 2 = -4$
   Point: $(-4, -4)$

2. $x = -3$:
   $y = -|-3 + 2| - 2 = -| -1 | - 2 = -1 - 2 = -3$
   Point: $(-3, -3)$

3. $x = -2$:
   $y = -|-2 + 2| - 2 = -| 0 | - 2 = 0 - 2 = -2$
   Point: $(-2, -2)$

4. $x = -1$:
   $y = -|-1 + 2| - 2 = -| 1 | - 2 = -1 - 2 = -3$
   Point: $(-1, -3)$

5. $x = 0$:
   $y = -|0 + 2| - 2 = -| 2 | - 2 = -2 - 2 = -4$
   Point: $(0, -4)$

Step 3: Plot the Points

You can now plot the points $(-4, -4)$, $(-3, -3)$, $(-2, -2)$, $(-1, -3)$, and $(0, -4)$ on the graph. Connect the points to form an upside-down V shape.

Step 4: Finalize the Graph

The vertex of the graph is at $(-2, -2)$, and the graph is symmetric about this vertex. Extend the graph further if needed.

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Related Questions:

1. How does the negative sign outside the absolute value affect the graph?
2. What happens to the graph if the $+2$ inside the absolute value is changed to $-2$?
3. What is the significance of the vertex in this function?
4. How can you find the domain and range of this function?
5. What is the general form of an absolute value function, and how can you transform it?

Tip:

When dealing with transformations, always work step by step: reflect, shift, and stretch/compress as necessary!