https://file.aiquickdraw.com/mathbot/file/1726562234214hjwkdy8y.jpg

The graph provided shows the function \( g(x) \), which is a transformation of the function \( f(x) = |x| \).

The graph appears to have the following transformations:

1. **Reflection**: The graph reflects the absolute value function over the x-axis to the left of \( x = 0 \) (i.e., for \( x < 0 \)).
2. **Vertical Translation**: The vertex of the graph is moved downward to \( (0, -3) \).
3. **Shift**: The absolute value function seems to be unchanged in terms of stretch or compression, but shifted downward by 3 units.

Thus, the transformation can be described as a vertical shift downward by 3 units and possibly a reflection of the left side of the function.

The function \( g(x) \) can be written as:
\[
g(x) = -|x| - 3
\]
This means the graph has undergone:
- A vertical reflection (for negative values of \( x \)).
- A downward shift of 3 units.

Would you like further explanation on how to derive the transformation, or do you have any specific questions?

Here are 5 related questions:
1. How do you determine reflections in function transformations?
2. What effect does a vertical shift have on a graph?
3. How would the function change if the graph were shifted 2 units to the right instead?
4. What does it mean when a function is reflected over the x-axis?
5. Can you apply multiple transformations, such as both horizontal and vertical shifts?

**Tip:** When shifting a function vertically, the transformation is directly applied to the output of the function, which is why you see constants added or subtracted outside the absolute value term.

The function g(x), graphed below, is a transformation of the function f(x) = |x|. Write the formula for g(x).

This problem involves determining the transformation of the absolute value function f(x) = |x|. The transformed function g(x) = -|x| - 3 shows a reflection over the x-axis and a vertical shift downward by 3 units. Suitable for Grades 9-12 students learning about function transformations.

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