In this image, you're asked to describe a piecewise function based on the graph. The graph shows three segments. Let’s break down each segment step by step to write the corresponding piecewise function.

Segment 1 (left side):

• Interval: This segment starts from $x = -4$ and ends at $x = -2$.
• Shape: It appears to be a decreasing line segment.
• Endpoints:
  • At $x = -4$, $y = 4$.
  • At $x = -2$, $y = 0$.
• Slope calculation: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 4}{-2 - (-4)} = \frac{-4}{2} = -2$
• Equation: Using the point-slope form $y - y_1 = m(x - x_1)$, pick the point $(-2, 0)$: $y - 0 = -2(x - (-2)) \implies y = -2(x + 2)$ Simplifying: $y = -2x - 4$

Segment 2 (middle):

• Interval: This segment starts from $x = -2$ and ends at $x = 1$.
• Shape: It's a horizontal line.
• Value: The line is constant at $y = 0$ for all $x$ in this interval.
• Equation: $y = 0$

Segment 3 (right side):

• Interval: This segment starts from $x = 1$ and ends at $x = 4$.
• Shape: It appears to be an increasing line segment.
• Endpoints:
  • At $x = 1$, $y = 0$.
  • At $x = 4$, $y = 3$.
• Slope calculation: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{4 - 1} = \frac{3}{3} = 1$
• Equation: Using the point-slope form with point $(1, 0)$: $y - 0 = 1(x - 1) \implies y = x - 1$

Piecewise Function:

Now, we can combine all the segments to write the piecewise function:

\begin{cases} -2x - 4 & \text{if } -4 \leq x \leq -2 \\ 0 & \text{if } -2 \leq x \leq 1 \\ x - 1 & \text{if } 1 \leq x \leq 4 \end{cases}$$ This is the piecewise function that represents the graph. If you have any further questions or need additional explanations, feel free to ask! Here are 5 related questions to consider: 1. How would the function change if the middle segment were sloped instead of horizontal? 2. What are the key differences between continuous and discontinuous piecewise functions? 3. How can you verify that the endpoints of a piecewise function match the graph accurately? 4. Can you express the same piecewise function using different forms, such as inequalities? 5. What happens to the graph if you shift one of the segments vertically? **Tip:** When dealing with piecewise functions, always double-check the slopes and y-intercepts to ensure the equations align with the graph segments.