https://file.aiquickdraw.com/mathbot/file/17338014546525p5rwqfp.jpg

To determine which graph matches each function, let's understand the transformations:

1. **\(y = |x|\)**  is a V-shaped graph opening upwards.
2. **\(y = -|x|\)** is a V-shaped graph opening downwards.
3. **\(y = -4|x|\)** is a vertically compressed and downward-opening V due to the negative and the factor 4.
4. **\(y = -4|x - 3|\)** is similar to the previous, but horizontally shifted 3 units to the right.

Now, let's analyze the given graphs:

- The graph in the **first option (top-left)** matches:
  - (a) \(y = |x|\) as the blue graph opening upwards.
  - (b) \(y = -|x|\) as the red graph opening downwards.
  - (c) \(y = -4|x|\) as the purple steeper graph opening downwards.
  - (d) \(y = -4|x - 3|\) as the green graph showing a rightward shift.

So, the **first option (top-left)** is the correct match.

Graph the functions y = |x|, y = -|x|, y = -4|x|, and y = -4|x - 3| on the same screen and identify which graph corresponds to each function.

Learn how to graph the absolute value functions y = |x|, y = -|x|, y = -4|x|, and y = -4|x - 3| on the same screen. This problem explores transformations, including reflections, vertical stretching, and horizontal shifts. Suitable for high school students in Grades 9-11.

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