https://file.aiquickdraw.com/mathbot/file/17266178664417idvob3r.jpg

The graph in your image shows two distinct parts, which suggest a piecewise function. Let's break it down to express this function:

1. **Left segment**: This appears to be a horizontal line from \( x = -9 \) to \( x = 2 \). The line has a constant value at \( y = 0 \), except at \( x = 2 \), where there's an open circle, meaning the function does not include the point \( (2, 0) \).
   
2. **Right segment**: Starting at \( x = 2 \), the graph shows a line with a positive slope, moving upwards. The segment begins at \( (2, -3) \) and passes through points like \( (4, 0) \) and \( (6, 3) \), indicating a linear relationship.

### Equation for the Piecewise Function:
1. For \( x \in [-9, 2) \), the function is constant at \( y = 0 \). Hence:
   \[
   f(x) = 0 \quad \text{for} \quad -9 \leq x < 2
   \]
   
2. For \( x \geq 2 \), the function follows a linear equation. To find the slope, use the points \( (2, -3) \) and \( (6, 3) \). The slope \( m \) is calculated as:
   \[
   m = \frac{3 - (-3)}{6 - 2} = \frac{6}{4} = \frac{3}{2}
   \]
   Now, use the point-slope form of a line:
   \[
   y - (-3) = \frac{3}{2}(x - 2)
   \]
   Simplifying this gives:
   \[
   f(x) = \frac{3}{2}(x - 2) - 3 \quad \text{for} \quad x \geq 2
   \]

### Complete Piecewise Function:
\[
f(x) =
\begin{cases}
0 & \text{for} \quad -9 \leq x < 2 \\
\frac{3}{2}(x - 2) - 3 & \text{for} \quad x \geq 2
\end{cases}
\]

Would you like more details or have any questions? 

Here are 5 related questions to explore:
1. How do you find the slope of a line between two points?
2. What is the significance of the open circle at \( x = 2 \)?
3. How can you graph a piecewise function from its algebraic expression?
4. What other types of functions can be expressed as piecewise functions?
5. How do we handle discontinuities in piecewise functions?

**Tip**: Always double-check which points are included in a piecewise function by looking for open or closed circles on the graph!

Express the function graphed on the axes as a piecewise function.

Learn how to express a function from a graph as a piecewise function. This problem involves identifying two linear segments, calculating the slope, and writing a piecewise function. Suitable for high school students in Grades 9-12.

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