AP Precalculus 3-Day Cram Plan

You have three days until the AP Precalculus exam. This is not the time to learn new material from scratch — it's time to drill the highest-frequency topics and lock in the FRQ patterns the College Board tests relentlessly.

This plan assumes ~4 focused hours per day. The exam has 4 FRQs and 40 multiple-choice questions spread across 4 units. Skip nothing on the checklist; if you're short on time, shorten the practice sets, not the topic coverage.

Day 1: Polynomial & Rational Functions + Exponential Growth (4 hrs)

The first 20–25 multiple-choice questions test these foundations hard. Make them automatic.

What to review (90 min)

• Polynomial end behavior: Leading coefficient + degree rule. $f(x) = -3x^4 + \ldots$ → as $x \to \infty$, $f(x) \to -\infty$.
• Zeros and factors: If $r$ is a zero, $(x-r)$ is a factor. Multiplicity and graph touch/cross behavior.
• Rational functions: Vertical asymptotes (denominator = 0, numerator ≠ 0), horizontal asymptotes (compare degrees of num and denom), holes (common factors).
• Polynomial division: Know long division or synthetic division to reduce rational functions.
• Average rate of change: $\frac{f(b)-f(a)}{b-a}$. Appears on ~every exam.
• Exponential models: $f(t) = a \cdot b^t$ and $f(t) = a e^{kt}$. Growth vs decay (is $k$ positive or negative?).

What to practice (2.5 hrs)

• 15 multiple-choice on polynomial zeros, end behavior, and rational function asymptotes (no calculator).
• 1 full FRQ on rational function analysis (asymptotes, zeros, domain, one graph from scratch).

💡 Highest leverage: Asymptotes and end behavior show up in 3+ questions per exam. If a student can write the asymptote and sketch the end behavior, they're already in the 4-5 range.

Day 2: Logarithmic/Exponential Modeling + Trigonometric Graphs (4 hrs)

FRQ 2 and FRQ 3 live here. These two areas account for nearly half of all points.

What to review (90 min)

• Log properties: $\log_b(xy) = \log_b(x) + \log_b(y)$, $\log_b(x^n) = n \log_b(x)$, $\log_b(1/x) = -\log_b(x)$. Change of base: $\log_b(x) = \frac{\ln(x)}{\ln(b)}$.
• Solving exponential and log equations: $5 \cdot 2^x = 80$ (isolate, take log of both sides). $\ln(x+3) = 2$ (exponentiate both sides).
• Sinusoidal graphs: $f(x) = a\sin(b(x-c)) + d$. Know what each parameter does:
  • $a$ = amplitude (vertical stretch).
  • $b$ = frequency (period = $2\pi / b$).
  • $c$ = phase shift (horizontal; note the minus sign).
  • $d$ = midline (vertical shift).
• Inverse trig functions: $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ and their ranges. Don't confuse with reciprocals ($\csc$, $\sec$, $\cot$).
• Trig identities: $\sin^2(\theta) + \cos^2(\theta) = 1$. Pythagorean, cofunction, angle sum (if needed).

What to practice (2.5 hrs — timed)

• 1 timed FRQ on sinusoidal modeling: given real-world context (temperature, light intensity, sound, etc.), write $f(x) = a\sin(b(x-c)) + d$ with correct period and phase shift.
• 1 FRQ on exponential or logarithmic modeling: interpret growth rate, make predictions, solve for time.
• 15 mixed multiple-choice on trig graph transformations and log/exp equations (no calculator).

⚠️ FRQ trap: When asked to write a sinusoidal model, most students forget the phase shift. The formula is $f(x) = a\sin(b(x-c)) + d$, not $a\sin(bx-c) + d$. The $c$ is a shift of the input, not a shift inside the argument.

Day 3: Polar Equations + Parametric/Vectors/Matrices + Full Mock (4 hrs)

What to review (90 min)

• Polar coordinates: $x = r\cos(\theta)$, $y = r\sin(\theta)$. Convert $r^2 = x^2 + y^2$, $\tan(\theta) = y/x$.
• Polar equations: $r = 2\cos(\theta)$ is a circle. Symmetry tests (polar axis, pole, line $\theta = \pi/2$).
• Parametric equations: Eliminate the parameter to get a Cartesian equation. If $x = t^2$ and $y = 2t+1$, solve for $t$ and substitute.
• Vectors: Magnitude $\|{\bf v}\| = \sqrt{v_1^2 + v_2^2}$. Dot product ${\bf u} \cdot {\bf v} = u_1 v_1 + u_2 v_2$. Angle between vectors.
• Matrix operations: Multiplication, inverses (2×2: $A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}$). Solving $A{\bf x} = {\bf b}$ via ${\bf x} = A^{-1}{\bf b}$.
• Transformations as matrices: Rotation, reflection, scaling represented as matrix multiplication.

What to practice (2.5 hrs — full timed set)

• 1 full FRQ on parametric to rectangular conversion or polar graph/equation.
• 1 FRQ on vectors or matrices (system of equations, transformations).
• 25 mixed multiple-choice (calculator + non-calculator), strictly timed.

The night before

Skim our last-minute review checklist. Get 8 hours of sleep — short-term memory consolidation is real, and a tired brain confuses $\sin(b(x-c))$ with $\sin(bx-c)$.

Calculator must-knows

On the calculator section, these functions are your friends:

1. Graph mode: Plot $y = 2\sin(3(x-\pi/4)) + 1$ and read key features (max, min, zeros).
2. Solver: Solve $2^x = 50$ numerically.
3. Matrix operations: Enter and multiply matrices, compute inverses.
4. Intersect function: Find where two graphs cross (useful for modeling verification).

Practice these buttons tonight, not for the first time on exam day.

Common point-leaks

• Forgetting the phase shift outside the sine argument.
• Confusing $\tan^{-1}$ with $1/\tan$ (reciprocal vs inverse).
• Forgetting to check domain restrictions on inverse trig (e.g., $\sin^{-1}$ only defined on $[-1, 1]$).
• Missing the negative sign in $\cos^{-1}$ derivative or writing asymptotes as $x = c$ instead of identifying vertical vs horizontal.
• Dropping signs on polar conversion (which quadrant is the point in?).
• Forgetting "+ C" on antiderivatives (less common in Precalc but still tested).

What this 3-day plan deliberately skips

You will not fully remaster every trig identity or matrix determinant in 3 days. If you're shaky on Cramer's rule or sum-to-product identities: skim the formula, do 2 example problems, and accept you may lose 2-3 points. Spend the saved time mastering sinusoidal modeling and asymptotes instead.

Ready to start?

Open the AP Precalculus topic library → and start with Day 1 polynomial end behavior. Mix in 3-5 practice problems per topic from the worked examples. You've trained for this. Now execute. 🎯

Next steps: After Day 3, review the FRQ practice guide to lock in model-writing templates.