Mastering Piecewise Functions: A Guide to Graphing

The Step-by-Step Graphing Process

Graphing a piecewise function may seem intimidating, but if you break it down into individual pieces, it becomes a simple matter of plotting points.

Step 1: Identify the Intervals

Look at the function definition to find your "fences." For example, if a function is defined as:

$$f(x) = \begin{cases} 4x, & 0 \le x \le 4 \\ 2(x-4) + 16, & x > 4 \end{cases}$$

The first "neighborhood" exists from $x=0$ to $x=4$, and the second "neighborhood" begins immediately after $x=4$.

Step 2: Create a Table of Values

For each piece of the function, create a separate table of values. You should pick $x$-values that fall within that specific piece's domain.

Exam Tip: Always include the boundary value (the "fence" value) in your table of values for every piece, even if the inequality is "strictly greater than" ($>$) or "strictly less than" ($<$). This allows you to know exactly where to place your open or closed circle.

Step 3: Plot and Connect

Using your table, plot the points for the first piece and connect them, making sure to stop at the boundary. Then, repeat the process for the next piece.

Working with Different Function Types

Piecewise functions are not limited to one shape. They can be composed of various mathematical models, including linear functions and quadratic functions.

Linear Pieces

A linear piece follows the form $y = mx + b$. These appear as straight lines on your graph. In the parking garage example, the first four hours are represented by the linear equation $y = 4x$.

Quadratic Pieces

A piecewise function can also transition from a line to a curve. For instance, a function might use a quadratic equation (like $y = x^2$) for one part of its domain. This results in a parabolic shape for that specific interval.

Summary Checklist

When you finish your graph, perform a quick check using these criteria:

1. Did I identify the "fence"? Ensure each function stays in its assigned $x$-interval.
2. Is the domain correct? Check that your pieces cover the intended $x$-values.
3. Are my points correct? Use your table of values to ensure accuracy.
4. Did I use open and closed circles? Verify that $\le$ or $\ge$ gets a closed circle and $<$ or $>$ gets an open circle.