https://file.aiquickdraw.com/mathbot/file/1726222227029z49vegnw.jpg

The graph provided represents a transformation of the function \( f(x) = |x| \), which is the absolute value function. To analyze this transformation:

1. The basic graph of \( f(x) = |x| \) has a vertex at \( (0, 0) \), with the graph increasing for \( x \geq 0 \) and decreasing for \( x < 0 \).
2. In this graph, the vertex has been shifted to \( (-3, -3) \).
3. Additionally, the slope of the right side remains \( 1 \) (upward), and the left side has a slope of \( -1 \), similar to the original function.

This corresponds to a horizontal and vertical translation. The transformation of \( f(x) = |x| \) can be written as:
\[
g(x) = |x + 3| - 3
\]
Here:
- \( x + 3 \) shifts the graph to the left by 3 units.
- \( -3 \) moves the graph down by 3 units.

Thus, the equation of the transformed function is:
\[
g(x) = |x + 3| - 3
\]

Would you like more details on this transformation or have any further questions?

### Related Questions:
1. How do you determine the transformation from the graph of a basic function?
2. What is the effect of adding or subtracting constants inside the absolute value function?
3. How would the graph change if the function were \( g(x) = |x - 3| + 3 \)?
4. Can you describe how the slope of the graph changes before and after the vertex?
5. How would you reflect the graph of \( f(x) = |x| \) over the x-axis?

### Tip:
When analyzing transformations, focus on shifts (horizontal/vertical), stretches, compressions, and reflections, as these changes are reflected in the form of the equation!

The graph above is a transformation of the function f(x) = |x|. Write an equation for the function graphed above.

Understand the transformation of the absolute value function f(x) = |x| by analyzing the graph, which includes horizontal and vertical shifts. This solution covers translation concepts for absolute value graphs, suitable for Grades 8-10.

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