https://file.aiquickdraw.com/mathbot/file/17268535639810ddxuabn.jpg

The problem asks us to identify which piecewise function corresponds to the given graph. The function graph consists of three distinct pieces, each defined over specific intervals:

1. For \( x < -2 \), the function appears to be a line segment with a positive slope, indicating a linear function.
2. For \( -2 \leq x < 2 \), the function has a parabolic shape opening downwards, suggesting a quadratic function.
3. For \( x \geq 2 \), the function seems to decrease linearly, suggesting another linear function but with a negative slope.

Let's compare this behavior to the given options:

1. **Option A**: 
   \[
   f(x) = \begin{cases} 
   x + 6, & x < -4 \\
   x^2, & -4 \leq x < 4 \\
   6 - x, & x \geq 4 
   \end{cases}
   \]
   This does not match the intervals or the function shapes.

2. **Option B**: 
   \[
   g(x) = \begin{cases} 
   x + 6, & x < -2 \\
   -x^2, & -2 \leq x < 2 \\
   6 - x, & x \geq 2 
   \end{cases}
   \]
   This matches the graph’s shape: linear in the first interval, quadratic (downward) in the second, and linear with a negative slope in the third interval.

3. **Option C**: 
   \[
   h(x) = \begin{cases} 
   x + 4, & x < -2 \\
   -x^2, & -2 \leq x < 2 \\
   4 - x, & x \geq 2 
   \end{cases}
   \]
   This option also has the correct general shape for each interval but does not match the exact line segments shown in the graph.

Based on the function segments and their behaviors, **Option B** is the correct match for the graph.

Would you like further details on why Option B is correct? Here are some related questions:

1. How do you determine the equation of a linear function from its graph?
2. What are the characteristics of a quadratic function that opens downward?
3. How do piecewise functions behave at their boundary points?
4. How can you verify a piecewise function by evaluating key points?
5. How does changing the coefficients of a function affect its graph?

**Tip:** When working with piecewise functions, always check the boundaries and make sure each segment matches the graph’s behavior over the specified interval.

Which rule describes the function whose graph is shown?

Learn how to identify a piecewise function from its graph, including linear and quadratic components. This problem is suitable for high school students in Grades 9-12.

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