Let's analyze the given graph and the functions:

1. Vertex: The graph has a V-shape, typical of absolute value functions, and the vertex is located at $(1, -2)$.
2. Slope: The slopes of the lines are positive and identical on both sides of the vertex. The graph appears to rise with a slope of 3 on either side.

We are looking for a function in the form $f(x) = a|x-h|+k$, where:

• $h$ is the x-coordinate of the vertex (shift),
• $k$ is the y-coordinate of the vertex (vertical shift),
• $a$ controls the steepness of the slope.

Identifying the correct function:

• The vertex is at $(1, -2)$, which means $h = 1$ and $k = -2$. This tells us that the absolute value should be of the form $|x - 1|$.
• The slope is 3, so the coefficient of the absolute value is 3.

Matching this to the given options:

• The function $f(x) = 3|x - 1| - 2$ fits these conditions, with a vertex at $(1, -2)$ and a slope of 3.

Correct Answer:

$f(x) = 3|x - 1| - 2$

Would you like further details on how absolute value functions work or the transformations involved?

Here are 5 related questions for further exploration:

1. What does the coefficient inside the absolute value function control?
2. How would the graph look if the coefficient of the absolute value were negative?
3. How does shifting the graph horizontally affect the equation of the function?
4. What changes would you make to the function to move the vertex to $(2, 3)$?
5. How do you calculate the slope from the graph of an absolute value function?

Tip: The vertex form $f(x) = a|x-h| + k$ is a powerful way to analyze transformations, with $h$ and $k$ representing horizontal and vertical shifts respectively.