Common Mistakes

Frequent Errors

1. Diagnosing exponential data as linear because it "goes up." 2. Writing $f(x)=a+rx$ for a geometric/exponential pattern. 3. Using sequence domain language for a continuous model (or vice versa). 4. Wrong point-slope: $y_i+m(x+x_i)$ instead of $x-x_i$. 5. Wrong point-ratio: $y_i r^{x+x_i}$ or $y_i r^{x/x_i}$. 6. Thinking many points are required when two distinct values determine the model (ideal case). 7. Confusing initial value $b$ or $a$ with the change constant $m$ or growth factor. 8. Treating discrete sequence graphs and continuous function graphs as the same picture.

Why These Happen

Students carry sequence formulas forward without adjusting domain or structure words (add vs multiply).

How to Fix Them

• Always test both differences and ratios on a table.
• Translation card: $(a_0,d)->(b,m)$ and $(g_0,r)->(a,b)$.
• Say domain out loud when switching representations.
• Rebuild from two points and verify a third if available.

Avoidance Scripts

• "Constant add => linear; constant multiply => exponential."
• "Sequences: whole numbers. Functions here: often all reals."
• "Two points can lock $b+mx$ or $ab^x$."

Quick Self-Check

• Did I verify differences vs ratios?
• Is my formula additive or multiplicative as intended?
• Did I address domain if comparing to a sequence?