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!