https://file.aiquickdraw.com/mathbot/file/1726145315685p2jou9k2.jpg

I see that you've uploaded an image showing graphs of three different functions, along with their equations. The task seems to involve sketching the graphs of the given functions and verifying them using a graphing utility.

Here are the given functions:

1. \( f(x) = |x| \) (Absolute value function)
2. \( g(x) = |4x| \) (Absolute value function with a coefficient of 4)
3. \( h(x) = -4|x + 4| - 5 \) (A transformed absolute value function)

### Explanation of the functions and their graphs:

1. **\( f(x) = |x| \)**:  
   This is a basic absolute value function. The graph is a V-shape, symmetric about the y-axis, with the vertex at the origin (0,0). The function is increasing for \( x > 0 \) and decreasing for \( x < 0 \).

2. **\( g(x) = |4x| \)**:  
   This graph is also a V-shape but steeper compared to \( f(x) \). Since the coefficient inside the absolute value is 4, the slope of the lines on either side of the vertex is 4 (positive on the right side and negative on the left side). The vertex remains at (0,0).

3. **\( h(x) = -4|x + 4| - 5 \)**:  
   This function involves multiple transformations of the absolute value function:
   - The term \( |x + 4| \) translates the graph 4 units to the left.
   - The coefficient \(-4\) reflects the graph over the x-axis and stretches it vertically by a factor of 4.
   - The \(-5\) translates the graph 5 units downward.
   Therefore, the vertex of this graph is at \( (-4, -5) \), and the V-shape opens downward.

The graph displayed confirms the transformation and behavior of these functions. The black graph represents \( h(x) \), the red graph represents \( g(x) \), and the blue graph represents \( f(x) \).

Would you like a deeper analysis of any specific function, or do you have any questions?

### 5 Related Questions:
1. What are the effects of vertical stretches and compressions on absolute value functions?
2. How do horizontal and vertical translations affect the graph of a function?
3. How does changing the sign of a coefficient impact the direction in which a graph opens?
4. Can we generalize transformations for other types of functions, like quadratic or exponential?
5. What happens if we introduce additional transformations, like reflections over the y-axis?

### Tip:
When graphing functions with absolute values, always consider how each transformation (shifts, stretches, compressions) affects the symmetry and orientation of the V-shape.

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility. Functions: f(x) = |x|, g(x) = |4x|, h(x) = -4|x + 4| - 5.

Learn how to graph absolute value functions and their transformations. Understand how to sketch f(x) = |x|, g(x) = |4x|, and h(x) = -4|x + 4| - 5 on the same coordinate system and verify results using a graphing utility. Suitable for high school students (Grades 9-12).

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