Exponential Functions — 700-800 - Complete Interactive Lesson

Part 1: The 700-800 Patterns

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 — not . 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 () 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 by recognizing , never by trial multiplication.

Part 2: Traps & Speed

Exponential Functions: Traps & Speed

Part 2 of 3 — Distractor Autopsy and Faster Routes

The Five Standard Distractors

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.
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). per two years is not per year.

Part 3: Timed Drill

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.

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 — 700-800