The function $f(x) = -4|x + 2| + 5$ is an absolute value function. Let's analyze it step by step.

1. Shape of the Graph:

• The function is of the form $f(x) = a|x - h| + k$, where $a = -4$, $h = -2$, and $k = 5$.
• The absolute value function $|x + 2|$ creates a "V" shape.
• The coefficient $a = -4$ indicates that the graph opens downwards (since $a$ is negative) and is vertically stretched by a factor of 4.

2. Vertex of the Function:

• The vertex of the graph is at $(h, k) = (-2, 5)$.
• This is the highest point on the graph since the graph opens downward.

3. Domain and Range:

• Domain: The function is defined for all real numbers, so $\text{Domain} = (-\infty, \infty)$.
• Range: Since the graph opens downward and the vertex is at $( -2, 5 )$, the maximum value of the function is 5. Thus, the range is $(-\infty, 5]$.

4. Symmetry:

• The graph is symmetric with respect to the vertical line $x = -2$, which is the line passing through the vertex.

5. Transformations:

• The graph of the basic function $|x|$ is shifted 2 units to the left (because of the $x + 2$ inside the absolute value).
• It is then reflected over the x-axis (because of the negative sign) and stretched vertically by a factor of 4.
• Finally, it is shifted upward by 5 units.

Would you like more details on any of these points? Here are some related questions you might consider:

1. What is the vertex form of an absolute value function?
2. How does the value of $a$ affect the steepness and direction of the graph?
3. What are some other examples of transformations on the basic absolute value function?
4. How can you determine the range of an absolute value function?
5. How do you graph the function step by step?

Tip: Understanding how transformations affect the graph of basic functions is key to quickly sketching and analyzing more complex functions.