Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What feature suggests there are discontinuities in a graph?

1. Slopes only

2. Holes or vertical asymptotes

3. Horizontal lines only

4. Completely smooth curves

The correct answer is: Holes or vertical asymptotes

The presence of holes or vertical asymptotes in a graph is a clear indication of discontinuities. Holes occur when a function is not defined at a particular point, typically because a factor cancels out during simplification, leaving a point that is 'missing' from the graph. This indicates that there is a break in the function at that specific x-value. Vertical asymptotes, on the other hand, occur when the function approaches infinity as it nears a certain x-value. In these cases, the function does not settle at a particular y-value but rather diverges to extreme values, causing a gap or discontinuity in the graph. In contrast, the other options do not indicate discontinuities. Slopes alone describe the behavior of the function but do not directly signal interruptions in continuity. Horizontal lines suggest a constant value but do not imply any breaks or discontinuous behavior. Completely smooth curves, representing continuous functions, would indicate continuity without any interruptions.