Common Mistakes

Frequent Errors

1. Calling a pattern periodic when repeats are not over equal-length input intervals. 2. Stating period as $2k$ when $k$ already works. 3. Sketching cycles that slowly change height (not periodic). 4. Forgetting that one interval of width $k$ determines the whole function. 5. Misreading a table's repeat start. 6. Describing features in period 1 that "disappear" later. 7. Confusing period with the length of an increase-only interval. 8. Assuming radians/degrees are required before 3.2+ content.

Why These Happen

Students rush to trig vocabulary and skip the CED definition: smallest $k>0$ with $f(x+k)=f(x)$, built from verbal/graphical cycles.

How to Fix Them

• Write the definition of period before computing.
• Test candidate periods: does $k/2$ also work? If yes, $k$ is not minimal.
• Trace one landmark (peak) across several cycles.

Avoidance Scripts

• "Smallest positive $k$ with $f(x+k)=f(x)$."
• "One cycle, then copy."
• "Features in one period appear in every period."

Quick Self-Check

• Is my $k$ the smallest that works?
• Does my extended graph match the first cycle?
• Did I use equal-length intervals in the justification?