To write an equation for a piecewise linear function based on a graph, you need to determine the following:
-
Identify Intervals: Break the graph into distinct intervals or segments where the function has different linear behavior.
-
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.
-
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.
-
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 \(y = mx + b\).
-
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).
-
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.
1 answer