The graph shown in the image is a transformation of the function $f(x) = |x|$, which is a V-shaped graph with a vertex at the origin. Let's analyze the given transformation.

Observations:

• The original vertex of $f(x) = |x|$ is at the origin, but in this graph, the vertex has shifted.
• The vertex of this new graph appears to be at $(-2, -3)$, which indicates a horizontal shift by 2 units to the left and a vertical shift by 3 units down.
• The slope of the graph on both sides of the vertex is the same as $f(x) = |x|$, meaning there is no change in slope (no stretching or compression).

Transformation Rule:

The general transformation for an absolute value function is: $g(x) = a |x - h| + k$ Where:

• $h$ represents a horizontal shift (to the left if $h$ is negative),
• $k$ represents a vertical shift (down if $k$ is negative),
• $a$ affects the slope (since $a = 1$, there is no vertical stretch or compression).

Equation:

Here, the graph has shifted horizontally by 2 units to the left, so $h = -2$, and vertically by 3 units downward, so $k = -3$. Therefore, the equation is: $g(x) = |x + 2| - 3$

Would you like more details on any step, or have further questions?

Related questions:

1. What is the effect of changing the slope $a$ in an absolute value function?
2. How would the graph change if the transformation had a vertical shift upwards?
3. Can you identify the domain and range of the transformed function $g(x) = |x + 2| - 3$?
4. How does reflecting the graph of $f(x) = |x|$ across the x-axis affect the equation?
5. What are the key differences between horizontal and vertical transformations in graphs?

Tip:

Always start by identifying shifts (horizontal and vertical) when graphing transformations of functions like $|x|$.