https://file.aiquickdraw.com/mathbot/file/1729818001330fixpv570.jpg

Let's analyze the graph and the piecewise functions to determine which one matches.

### Observations from the graph:
1. **For \( x \leq -2 \)**: The function appears to be linear, with a downward slope, indicating that the function could be of the form \( f(x) = -2x - 3 \).
2. **For \( -2 < x < 2 \)**: The function shows a parabolic behavior, opening upwards. This suggests the function is quadratic, likely of the form \( f(x) = x^2 \) or similar.
3. **For \( x \geq 2 \)**: The graph becomes constant at \( y = 2 \), indicating the function is constant for this interval.

### Analyze each option:

#### Option A:
\[
f(x) = 
\begin{cases} 
-2x - 3 & x \leq -2 \\
x^2 + 2 & -2 < x < 2 \\
2^x & x \geq 2
\end{cases}
\]
- The first part matches the linear behavior for \( x \leq -2 \).
- The second part \( x^2 + 2 \) seems incorrect because the graph intersects the y-axis at 0, not 2.
- The third part \( 2^x \) is an exponential function, but the graph is constant at \( y = 2 \), so this does not match.

#### Option B:
\[
f(x) = 
\begin{cases} 
-2x - 3 & x < -2 \\
x^2 & -2 \leq x \leq 2 \\
2 & x > 2
\end{cases}
\]
- The first part matches the linear segment for \( x < -2 \).
- The second part \( x^2 \) matches the parabolic shape between \( -2 \leq x \leq 2 \).
- The third part \( 2 \) matches the constant part for \( x > 2 \).

#### Option C:
\[
f(x) = 
\begin{cases} 
-2x + 3 & x < -2 \\
x^2 - 2 & -2 \leq x \leq 2 \\
2 & x > 2
\end{cases}
\]
- The first part \( -2x + 3 \) does not match the linear behavior of the graph.
- The second part \( x^2 - 2 \) does not match because the graph reaches 0 at \( x = 0 \), not \( -2 \).
- The third part \( 2 \) is correct, but overall, this option does not match.

#### Option D:
\[
f(x) = 
\begin{cases} 
-2x - 3 & x < -2 \\
-x^2 + 2 & -2 \leq x \leq 2 \\
2^x & x > 2
\end{cases}
\]
- The first part \( -2x - 3 \) is correct.
- The second part \( -x^2 + 2 \) does not match the parabolic curve because the parabola opens upwards, not downwards.
- The third part \( 2^x \) does not match the constant \( y = 2 \) in the graph.

### Conclusion:
The correct piecewise function that matches the graph is **Option B**.

Would you like any further details or explanations? 

### Related Questions:
1. How do you determine the linear function from a graph?
2. What are the properties of quadratic functions and how can you recognize them on a graph?
3. How can you identify when a function is constant on a graph?
4. How do exponential functions compare to constant functions visually?
5. What are the general steps to analyzing a piecewise function from a graph?

**Tip:** When matching a piecewise function to a graph, break down the graph into different intervals and focus on recognizing the patterns (linear, quadratic, constant, etc.) within each segment.

Which of the following piecewise functions is graphed above?

Learn how to match a graph with its corresponding piecewise function consisting of linear, quadratic, and constant parts. This example covers key algebraic concepts such as recognizing function behaviors and analyzing graph patterns. Suitable for high school students in Grades 9-12.

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