Question

What information do I need to determine in order to write the equation for a piecewise linear function shown on a graph?

Answer 1

To write the equation for a piecewise linear function based on a graph, you need to determine the following information:

1. **Identify the pieces**: Look at the graph and determine how many distinct linear segments (pieces) there are. Each segment will have its own linear equation.

2. **Determine the endpoints**: For each linear segment, identify the x-coordinates (and corresponding y-coordinates) at the endpoints of the segment. This will help you understand the domain of each piece.

3. **Calculate the slope**: For each linear segment, calculate the slope (m) using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where (x_1, y_1) and (x_2, y_2) are the coordinates of the two endpoints of the segment.

4. **Find the y-intercept (or use point-slope form)**: You can either find the y-intercept (b) by looking at where the line crosses the y-axis, or you can use the point-slope form of the linear equation:
\[
y - y_1 = m(x - x_1)
\]
to express the equation of the line in slope-intercept form, \(y = mx + b\).

5. **Define the domain for each piece**: Specify the x-values (intervals) for which each linear equation is valid, based on the endpoints identified earlier. This typically involves using inequality notation (e.g., \( a \leq x < b \)) to define the domain for each piece.

After you have gathered this information, you can combine the equations and their corresponding domains into a piecewise function notation, usually in the form:

\[
f(x) =
\begin{cases}
\text{equation for first segment} & \text{for } x \text{ in interval 1} \\
\text{equation for second segment} & \text{for } x \text{ in interval 2} \\
\vdots & \vdots \\
\end{cases}
\]

By following these steps, you will be able to write the complete piecewise linear function that corresponds to the graph.