Common Mistakes

Frequent Errors

1. Using $b=1$ or $b\le 0$ in standard form. 2. Identifying a vertex/min/max on the entire real domain. 3. Claiming an inflection point on an exponential graph. 4. Confusing $a$ with the base $b$. 5. Saying decay means negative $a$ only (decay is $0<b<1$ with $a>0$). 6. Forgetting domain is all real numbers. 7. Wrong limit direction ($x\to\infty$ vs $x\to-\infty$). 8. Treating $f(x)=2^{x+1}$ as non-exponential.

Why These Happen

Students import quadratic vertex language and linear slope ideas onto exponentials.

How to Fix Them

• Checklist: $a$, $b$, $b$ vs 1, $f(0)$, ratio test.
• Limit template for growth and decay bases.
• Say: "always inc or dec; always concave one way; no inflection on $\mathbb{R}$."

Avoidance Scripts

• "$f(0)=a$."
• "$b>1$ grow; $0<b<1$ decay."
• "No turning point on all reals."

Quick Self-Check

• Did I state $a$ and $b$ correctly?
• Growth or decay?
• Limits for both directions?
• Extrema/inflection only if a closed interval is given?