The given graph is a transformation of the absolute value function $f(x) = |x|$.

Key observations:

1. The graph is a "V" shape, which is characteristic of an absolute value function.
2. The vertex of the graph is at the point $(-1, 3)$, indicating that the graph has been shifted.
3. The graph opens downward, which means there is a reflection over the x-axis (indicating a negative sign).
4. The slopes of the lines are different: the slope on the left of the vertex is $+1$ (as usual), and on the right, it is $-1$, suggesting the graph behaves linearly on both sides, consistent with absolute value functions.

General transformation formula:

The absolute value function $f(x) = |x|$ can be transformed as: $g(x) = a|x - h| + k$ Where:

• $(h, k)$ is the vertex.
• $a$ represents a vertical stretch/compression or reflection.

Applying the transformations:

• The vertex is at $(-1, 3)$, so $h = -1$ and $k = 3$.
• The graph is reflected over the x-axis, so $a = -1$.

Thus, the equation for the transformed function is: $g(x) = -|x + 1| + 3$

Would you like more details or have any questions?

Related questions:

1. What other types of transformations can be applied to the absolute value function?
2. How would the formula change if the graph had been vertically stretched?
3. How do horizontal shifts affect the formula of an absolute value function?
4. What happens to the graph of an absolute value function if only the value of $a$ is changed?
5. Can you derive the formula for a graph with multiple transformations (e.g., vertical shift and stretch)?

Tip:

Always identify the vertex and slope when transforming absolute value functions—these are key to forming the correct equation.