Graphing Piecewise Functions with Boundary Points
Graphing piecewise functions with boundary points is a Grade 11 algebra skill in Big Ideas Math. A piecewise function uses different expressions over different intervals of the domain. To graph one, evaluate each piece only over its specified interval, use open circles (hollow) at endpoints excluded by strict inequalities (< or >), and closed circles (filled) at endpoints included by ≤ or ≥. For example, f(x) = x + 2 for x < 0 and f(x) = x² for x ≥ 0 creates a line for negative x values and a parabola for non-negative values, meeting at x = 0 with an open circle on the line piece and closed circle on the parabola piece.
Key Concepts
When graphing piecewise functions, use closed circles (•) to indicate points included in the domain of a piece, and open circles (○) to indicate points not included. At boundary points where domain intervals meet, only one piece can contain the boundary value.
Common Questions
What is a piecewise function?
A piecewise function uses different formulas over different parts (intervals) of its domain. Each 'piece' applies only to the specified x-values.
How do you determine whether to use an open or closed circle at a boundary point?
Use an open circle when the endpoint is excluded (strict inequality < or >). Use a closed circle when the endpoint is included (≤ or ≥).
How do you graph each piece of a piecewise function?
For each piece, treat it as its own function but plot only the portion within its defined interval. Apply boundary circle rules at the endpoints of each interval.
What does it mean when two pieces of a piecewise function share a boundary point?
The function value at the boundary is defined by whichever piece includes that point (closed circle). If neither includes it, the function is undefined at that exact point.
How do you evaluate a piecewise function at a specific x-value?
Determine which interval contains x, then substitute x into the corresponding formula for that interval. Only one formula applies to each x-value.
What is the domain and range of a piecewise function?
The domain is the union of all intervals specified. The range is the set of all output values from all pieces combined, found by analyzing each piece over its interval.