AP Precalculus 7-Day Cram Plan

title: "AP Precalculus 7-Day Cram Plan" description: "A structured week-long AP Precalculus study plan: daily focus areas, cumulative review, 2 full mock exams, and strategic FRQ practice to hit a 4-5." date: "2026-01-15" examDate: "May AP Exam" topics:

• Polynomial Functions
• Rational Functions
• Exponential & Logarithmic Functions
• Trigonometric Functions
• Polar & Parametric
• Vectors & Matrices

You have 7 days to review AP Precalculus. This plan gives you time to cover all four units and run two full practice exams with proper pacing and error analysis.

Commit to 3–4 hours per day, with one lighter day mid-week. The structure: focus on one major unit per day, then aggregate into full mocks on Days 6 and 7.

Daily Study Breakdown

| Day | Focus | Topics | Key Activity | Time | |---|---|---|---|---| | 1 | Polynomial Functions | End behavior, zeros, multiplicity, average rate of change | Review formulas; 20 MCQs + 1 graph-sketching exercise | 3.5 hrs | | 2 | Rational Functions | Asymptotes, holes, long division, function analysis | 15 no-calc MCQs + 1 FRQ on asymptote identification | 4 hrs | | 3 | Exponential & Log Functions | Growth/decay models, log properties, solving equations | 18 MCQs + 1 modeling FRQ (predict, interpret) | 3.5 hrs | | 4 | Trigonometric Functions | Unit circle, graphs, transformations, inverse trig | 25 MCQs (emphasis on graph transformations) | 4 hrs | | 5 | Sinusoidal Modeling + Identities | Writing $f(x) = a\sin(b(x-c)) + d$, solving trig equations | 2 full FRQ-style modeling problems (real-world context) | 3.5 hrs | | 6 | Parametric, Polar, Vectors | Parameter elimination, polar coordinates, vector operations | 20 MCQs + 1 FRQ on polar/parametric conversion | 4 hrs | | 7 | Mock Exam 1 + Matrices | Full 80-min timed exam, then review + matrix systems | Full-length timed test; error analysis for 2 hrs | 4 hrs |

Note: Day 7 includes one full practice exam; repeat on Day 8 if you have time.

Unit-by-unit milestones

Day 1: Polynomial Functions

• Know cold: End behavior (leading term only), factored form zeros, multiplicity (touch vs cross), turning points.
• Practice goal: Identify end behavior in 10 seconds. Sketch a polynomial from a factored form without a calculator.
• Red flag: If you're still struggling with "does degree 4 go up or down on the right," do 10 more end-behavior problems tonight.

Day 2: Rational Functions

• Know cold: Vertical asymptotes (set denom = 0, check numerator ≠ 0), horizontal asymptotes (compare degrees), slant asymptotes (polynomial long division if degree(num) = degree(denom) + 1).
• Practice goal: Write asymptotes in 30 seconds. Perform polynomial division to reduce a rational function.
• Trap: Confusing "hole" (removable discontinuity) with "vertical asymptote" (non-removable). A hole is where both numerator and denominator are zero.

Day 3: Exponential & Logarithmic

• Know cold: Log properties (product, quotient, power), change of base formula, solving $2^x = 50$ via logarithms, interpreting $e^{kt}$ as growth rate $k$.
• Practice goal: Solve $\log_3(x+1) = 2$ in your head. Convert $A = 1000 \cdot 1.05^t$ to the form $A = 1000 e^{kt}$ and identify $k$.
• Modeling checklist: Identify the initial amount $a$, the growth/decay factor $b$ or rate $k$, determine time variable, make predictions.

Day 4: Trigonometric Graphs

• Know cold: Unit circle (key angles 0, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$), basic graphs of $\sin$, $\cos$, $\tan$, ranges of inverse functions.
• Practice goal: Sketch $y = 2\cos(x - \pi/4) - 1$ without a calculator in 60 seconds. Identify amplitude, period, phase shift, and midline from a graph.
• Trap: $\cos^{-1}(x)$ has range $[0, \pi]$, not $[-\pi/2, \pi/2]$.

Day 5: Sinusoidal Modeling + Trig Identities

• Know cold: Given a real-world periodic scenario, identify amplitude = (max−min)/2, period (from "how long does the cycle take"), midline = (max+min)/2, phase shift (where does max or min occur?).
• Practice goal: Write a complete sinusoidal model from a word problem. Solve $\sin(2\theta) = 0.5$ for $\theta \in [0, 2\pi)$.
• FRQ habit: Always show your parameter reasoning. "Amplitude is 5 because the temperature ranges from 50 to 60 degrees" (midline = 55, amplitude = 5).

Day 6: Parametric, Polar, Vectors

• Know cold: Parametric-to-Cartesian (eliminate the parameter). Polar $\leftrightarrow$ Cartesian ($x = r\cos\theta$, $y = r\sin\theta$). Vector magnitude and dot product.
• Practice goal: Convert $x = 3\cos(t)$, $y = 2\sin(t)$ to $\frac{x^2}{9} + \frac{y^2}{4} = 1$ in 2 minutes. Convert $r = 4\cos(\theta)$ to Cartesian form.
• Trap: Remember the parametric domain; $t \in [0, 2\pi]$ traces a full ellipse.

Day 7 (Mock) + Matrices

• Complete mock exam: 40 MCQs (non-calc + calc) and 4 FRQs in 80 minutes. Time yourself.
• Error analysis: For each wrong answer, label whether it's computational error, conceptual misunderstanding, or misreading the question.
• Matrix systems: Solve $\begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}$ using $A^{-1}{\bf b}$ or row reduction.

Study tips by day

| Day | Study Strategy | |---|---| | 1–3 | Focus + quantity: 20+ MCQs per day, light FRQ. Build speed. | | 4–5 | Graphs matter: Spend time on the graphing calculator. Sketch by hand, then verify on calc. | | 6 | Mixed review: Jump between units. Mimic exam randomness. | | 7 | Timing discipline: Full timed mock; no breaks except a 5-minute stretch. |

Common mistakes by topic

• Polynomial/Rational: Forgetting vertical asymptote at a hole, misidentifying horizontal asymptote when degrees are equal.
• Exponential/Log: Confusing $\log(x^2) = 2\log(x)$ (only if $x > 0$; if $x < 0$, the left side is still defined but right is not).
• Trig: Phase shift sign error: $\sin(2(x - \pi/4)) = \sin(2x - \pi/2)$, so the shift is $+\pi/4$ to the right.
• Parametric/Polar: Forgetting to convert back to $(x, y)$ or $(r, \theta)$ in the final answer.
• Vectors/Matrices: Sign errors in matrix inverse or dot product.

🎯 Mid-week morale check: On Day 4, if you're feeling overwhelmed, pick your weakest unit and block out an extra hour tomorrow. Better to know 3 units cold than to skim all 4 poorly.

Before Day 7 (mock day)

• Print or pull up a full practice exam.
• Silence your phone. Set a timer.
• Mimic real exam conditions: no notes, calculator for Part B only, scratch paper.
• After you finish, do not check answers immediately. Let your brain rest 2 hours, then review.

Ready?

Jump to Day 1 → Polynomial end behavior. Spend 3 focused hours, then rest. See you tomorrow.

After the mocks: Review the FRQ practice guide to strengthen any unit where you scored below 60%.