5.2: Absolute Value Functions - Mathematics LibreTexts - Free Printable

It looks like you've uploaded an image containing three graphs labeled (a), (b), and (c), each showing a piecewise linear function on the coordinate plane. However, since I can't view images directly, I'll describe how to analyze such problems based on typical tasks associated with these types of graphs.

Let’s assume the task is to identify which graph represents a given function, or perhaps to determine the equation of each graph, or analyze transformations between them.

I’ll go through each graph and describe its features so we can solve common problems related to such graphs.

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Graph (a):

- The graph has a V-shape, symmetric about the y-axis.
- Vertex at (-1, 2).
- Left side: from (-4, 5) to (-1, 2)
- Right side: from (-1, 2) to (2, 5)
- Slope on left: (2 - 5)/(-1 + 4) = (-3)/3 = -1
- Slope on right: (5 - 2)/(2 + 1) = 3/3 = 1
- This resembles an absolute value function shifted.

Let’s write the equation:
Let $ f(x) = |x + 1| + 2 $

Check:
- At x = -1 → |0| + 2 = 2 ✔
- At x = 2 → |3| + 2 = 5 ✔
- At x = -4 → |-3| + 2 = 5 ✔

✔ So, Graph (a) is:
$$
f(x) = |x + 1| + 2
$$

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Graph (b):

- V-shaped, vertex at (2, 0)
- Left side: from (-1, 3) to (2, 0)
- Right side: from (2, 0) to (5, 3)
- Slope left: (0 - 3)/(2 + 1) = -3/3 = -1
- Slope right: (3 - 0)/(5 - 2) = 3/3 = 1

This is also an absolute value function, vertex at (2, 0)

So:
$$
f(x) = |x - 2|
$$

Check:
- x = 2 → |0| = 0 ✔
- x = 5 → |3| = 3 ✔
- x = -1 → |-3| = 3 ✔

✔ So, Graph (b) is:
$$
f(x) = |x - 2|
$$

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Graph (c):

- This is a triangle-like shape, not symmetric.
- Starts at (-4, -3), goes up to (1, 3), then down to (4, -1)
- Two line segments:
1. From (-4, -3) to (1, 3): slope = (3 + 3)/(1 + 4) = 6/5 = 1.2
2. From (1, 3) to (4, -1): slope = (-1 - 3)/(4 - 1) = -4/3 ≈ -1.33

Not an absolute value function — it's a piecewise linear function with a peak at (1, 3).

So, this is not an absolute value function.

We can define it as:
$$
f(x) =
\begin{cases}
\frac{6}{5}(x + 4) - 3 & \text{if } x \leq 1 \\
-\frac{4}{3}(x - 1) + 3 & \text{if } x > 1
\end{cases}
$$

But more simply, just note that it's a piecewise linear function with a maximum at (1, 3).

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Possible Task Interpretations:

#### Task 1: Which graph represents $ f(x) = |x - 2| $?
Answer: (b)

#### Task 2: Which graph represents $ f(x) = |x + 1| + 2 $?
Answer: (a)

#### Task 3: Identify the function for graph (c)
Answer: It's a piecewise linear function with two segments:
- Increasing from (-4, -3) to (1, 3)
- Decreasing from (1, 3) to (4, -1)

#### Task 4: Compare transformations
- (a) is $ |x| $ shifted left 1 and up 2
- (b) is $ |x| $ shifted right 2
- (c) is not a transformation of absolute value

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Final Answer (assuming the question is: "Which graph represents $ f(x) = |x - 2| $ ?")

👉 Answer: (b)

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If your actual task was different (e.g., find domain/range, symmetry, transformations, or equations), please clarify and I’ll adjust accordingly.

Let me know what the specific question is, and I’ll give a precise solution!