Common Mistakes

Frequent Errors

1. Period $=2\pi|b|$ or $|b|/(2\pi)$ instead of $2\pi/|b|$. 2. Calling the phase shift $+c$ for $y=\sin(b(\theta+c))+d$. 3. Using $a$ (signed) as amplitude without absolute value. 4. Shifting vertically by $a$ instead of $d$. 5. Forgetting reflection when $a<0$. 6. Mis-factoring $b\theta+k$ into $b(\theta+c)$. 7. Thinking cosine transforms differently from sine for $a,b,c,d$. 8. Phase-shifting by $c$ units in the wrong direction on a sketch.

Why These Happen

Students juggle multiple parameters and lose the CED dictionary: $d$ vertical, $-c$ phase, $|a|$ amplitude, $2\pi/|b|$ period (3.6.A.2–A.6).

How to Fix Them

• Write a parameter table before graphing.
• Always factor: $b\theta+k=b(\theta+k/b)$ so $c=k/b$ and phase $=-c=-k/b$.
• Check one landmark point after transforming.

Avoidance Scripts

• "Amplitude $|a|$; period $2\pi/|b|$; midline $d$; phase $-c$."
• "Factor $b$ before reading phase."
• "Negative $a$ reflects across the midline."

Quick Self-Check

• Is my period $2\pi$ divided by $|b|$?
• Is phase $-c$ for the $(\theta+c)$ form?
• Does midline match $d$?