Question

How would you explain to a classmate a general rule for writing piecewise functions for an
absolute value graph?

Answer 1

To write a piecewise function for an absolute value graph, you can generally follow this rule:

1. Identify the key points on the graph:
- The vertex: the point where the graph changes direction.
- The x-intercepts: the points where the graph intersects the x-axis.

2. Determine the behavior of the graph on different intervals:
- When x is less than the x-coordinate of the vertex, the graph is decreasing.
- When x is greater than the x-coordinate of the vertex, the graph is increasing.

3. Write the piecewise function using the identified points and behavior:
- Write the expression for the graph before the vertex, considering its decreasing behavior.
- Write the expression for the graph after the vertex, considering its increasing behavior.

For example, let's say the vertex of the absolute value graph is at (3, -2) and the x-intercepts are at (-1, 0) and (7, 0). The general rule for writing the piecewise function would be:

f(x) = {
-a(x - 3) - 2, when x ≤ 3,
a(x - 3) - 2, when x > 3,
}

Here, 'a' represents the slope of the graph before and after the vertex. The actual value of 'a' needs to be determined using the given key points.