To write the piecewise equations based on the graph you provided, I will analyze each segment of the graph. The graph contains three distinct line segments, each of which can be described by a linear equation in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Segment 1:

This segment is horizontal and located on the left side of the y-axis.

• Range: $-9 \leq x \leq 0$
• Equation: Since it's horizontal, the slope $m = 0$. Let's identify the y-value (which is constant): $y = -5$

Segment 2:

This segment is decreasing, crossing the x-axis.

• Range: $0 < x \leq 1$
• Slope Calculation: Pick two points on the line, say $(0, 0)$ and $(1, -5)$: $m = \frac{-5 - 0}{1 - 0} = -5$ Now, use point-slope form $y - y_1 = m(x - x_1)$: $y = -5x + 0 \implies y = -5x$

Segment 3:

This segment is increasing and extends to the right.

• Range: $1 < x \leq 4$
• Slope Calculation: Pick two points, say $(1, -5)$ and $(4, 10)$: $m = \frac{10 - (-5)}{4 - 1} = \frac{15}{3} = 5$ Use point-slope form $y - y_1 = m(x - x_1)$: $y + 5 = 5(x - 1)$ Simplifying: $y = 5x - 10$

Piecewise Function

Combining all these, the piecewise function is:

-5 & \text{for } -9 \leq x \leq 0, \\ -5x & \text{for } 0 < x \leq 1, \\ 5x - 10 & \text{for } 1 < x \leq 4. \end{cases}$$ Would you like more details or have any questions? ### Related Questions: 1. How would the piecewise function change if the slopes of the segments were different? 2. How do you calculate the slope of a line given two points? 3. How do you determine the y-intercept from the graph? 4. Can you explain how the domain of each piece of the piecewise function is determined? 5. What is the significance of the points where the graph changes direction? ### Tip: When dealing with piecewise functions, always carefully analyze the domain of each piece to ensure the correct function is applied within each interval.