Exponential Functions — 700-800

Exponential Functions: The 700-800 Patterns

Part 1 of 3 — The Archetypes Hard-Tier Items Are Built From

Hard exponential items are about growth factors and time units, not about plugging into formulas. Three archetypes dominate.

Archetype 1: Rebasing the Time Unit

A model like $A(t) = 60(0.5)^{t/3}$ ($t$ in hours) states its factor per 3 hours. To get the per-hour rate, rewrite:

$A(t) = 60\left[(0.5)^{1/3}\right]^{t}$

and $(0.5)^{1/3} \approx 0.7937$ — about a $20.6\%$ decrease per hour. The direction of the conversion is the whole trap: to shrink the time unit you take a root (divide the exponent's period), never a power.

Worked example. $V(y) = 340(1.15)^{y/2}$ grows $15\%$ per TWO years. Per year: $(1.15)^{1/2} \approx 1.0724$, so about $7.24\%$ — not $7.5\%$. Splitting a percent linearly is always slightly wrong, and the test puts the linear split among the choices.

Archetype 2: Compound vs. Linear

Two plans: one loses a fixed amount per year, one loses a fixed percent per year. Or: a nominal $6\%$ annual rate compounded monthly. The compound side must be computed as a power — $(1.005)^{12} \approx 1.0617$, so the effective rate is $6.17\%$, not $6\%$. Every linear shortcut (a monthly $0.5\%$ times $12$ is $6\%$) is a planted distractor; the true compound answer for decay is always LESS loss than the linear estimate, and for growth MORE gain.

Archetype 3: Counting Doublings and Halvings

"Doubles every 5 days, fills the lake at day 55 — when was it a quarter full?" Work in doublings, not amounts: a quarter is exactly two doublings before full, so day $45$. Half-life problems run the same way in reverse: $9$ grams at year $60$ with a $12$-year half-life means multiply by $2$ for every $12$ years you step back. Solve $2^{k} = 2048$ by recognizing $2048 = 2^{11}$, never by trial multiplication.

Exponential Functions: Traps & Speed

Part 2 of 3 — Distractor Autopsy and Faster Routes

The Five Standard Distractors

1. Percent remaining vs. percent decrease. If $64\%$ remains, the decrease is $36\%$. The complement of the right answer is ALWAYS a choice on decay items.
2. The linear split / linear stack. $2.5\%$ per month is not $30\%$ per year (it is about $26.2\%$, because each month's loss comes off a smaller base). $15\%$ per two years is not $7.5\%$ per year.
3. Conversion run backward. Rewriting $b^{t/4}$ requires the base $b^{1/4}$ (a root). Writing $(b^{4})^{t}$ multiplies the exponent instead of dividing — the reversed conversion is planted, usually as a suspiciously large percent.
4. Off-by-one in periods. "First exceeds," "first drops below": the last period BEFORE the crossing and the period AFTER it are both choices. Compute the boundary values and check which side of the threshold each sits on.
5. The intermediate amount. In half-life back-solving, the original amount and the answer to a neighboring time are planted.

Speed Techniques

• Powers of 2 recognition: $256, 512, 1024, 2048, 4096 = 2^{8}$ through $2^{12}$. Ratios in doubling problems are always a clean power of 2 — find the exponent, never multiply forward step by step.
• Step in half-lives, not years: "$24$ years is $3$ half-lives before the $60$-year measurement" turns a two-stage computation into one multiplication by $2^{3}$.
• Desmos exploit for thresholds: graph $y = 250(1.06)^{x}$ and $y = 400$ and click the intersection; the first integer to the right of it is the answer. This kills off-by-one errors, but still verify the integer on both sides.
• Compare-plans discipline: compute each plan fully, label each number, and reread which quantity (a value, a loss, or a difference) the question wants. All three of the other quantities will be among the choices.

Exponential Functions: Timed Drill

Part 3 of 3 — Four Questions at Full Difficulty

Budget about 75 seconds per question. Before computing anything, name the archetype: rebasing, compound-vs-linear, doubling count, or threshold crossing. Decide whether the answer should sit above or below the linear estimate — that alone often eliminates two choices. Use the calculator for powers like $(0.93)^{4}$, but set up the expression by hand first.