Question

In order to write an equation for a piecewise linear function shown on a graph, what do you need to determine?

Answer 1

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

1. **Identify Intervals**: Break the graph into distinct intervals or segments where the function has different linear behavior.

2. **Identify Points**: For each segment, determine the key points that define the endpoints and any transitions in the function. These points will typically be the vertices or intersections on the graph.

3. **Determine Slopes**: 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 any two points on that segment.

4. **Find y-Intercepts**: Once you have the slope, you can use one of the points on the segment to find the y-intercept (b) using the equation of a line \(y = mx + b\).

5. **Write the Equations**: For each segment, write the linear equation in the form \(y = mx + b\), specifying the appropriate interval for x (i.e., determining the domain for each piece).

6. **Combine into a Piecewise Function**: Finally, combine all of the individual piece equations into a piecewise function, clearly indicating the conditions (intervals) under which each equation applies.

By completing these steps, you will have a complete piecewise linear function that accurately reflects the behavior of the graph.