AP Precalculus Last-Minute Review (Night Before)

title: "AP Precalculus Last-Minute Review (Night Before)" description: "The night-before AP Precalculus checklist: must-know formulas, trig identities, log properties, common traps, parametric conversions, and morning-of strategy." date: "2026-01-15" examDate: "May AP Exam" topics:

• Formula Sheet
• Common Traps
• Calculator Tips

The exam is tomorrow. This is not the time to learn new content — it's time to skim formulas, reset your mind, and sleep. Spend 40–50 minutes on this page, then close your notes.

Polynomial & Rational Functions Quick Reference

| Concept | Formula / Rule | |---|---| | Polynomial end behavior | Leading term dominates: $-3x^4 + \ldots \to -\infty$ as $x \to \infty$ | | Zeros & factors | If $r$ is a zero, $(x-r)$ is a factor; multiplicity $n$ means graph touches (even $n$) or crosses (odd $n$) | | Vertical asymptote | Set denominator = 0, check numerator ≠ 0 | | Horizontal asymptote | Compare degrees: if $\deg(\text{num}) < \deg(\text{denom})$, then $y = 0$; if equal, $y = \frac{a_{\text{lead}}}{b_{\text{lead}}}$ | | Slant asymptote | If $\deg(\text{num}) = \deg(\text{denom}) + 1$, use polynomial long division | | Average rate of change | $\frac{f(b)-f(a)}{b-a}$ |

Exponential & Logarithmic Functions

| Type | Formula | Example | |---|---|---| | Exponential form 1 | $A(t) = a \cdot b^t$ | $A(t) = 1000 \cdot 1.05^t$ (5% growth per unit $t$) | | Exponential form 2 | $A(t) = a e^{kt}$ | $A(t) = 1000 e^{0.049t}$ (equivalent to $b = 1.05$ if $e^k = 1.05$) | | Relationship | $b = e^k$, so $k = \ln(b)$ | If $b = 1.05$, then $k = \ln(1.05) \approx 0.0488$ | | Log properties | $\log(xy) = \log(x) + \log(y)$; $\log(x^n) = n\log(x)$; $\log(1/x) = -\log(x)$ | $\log_2(8 \cdot 4) = \log_2(8) + \log_2(4) = 3 + 2 = 5$ | | Change of base | $\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log(x)}{\log(b)}$ | $\log_3(9) = \frac{\ln(9)}{\ln(3)} = \frac{2\ln(3)}{\ln(3)} = 2$ |

Trigonometric Functions & Unit Circle

| Angle | $\sin(\theta)$ | $\cos(\theta)$ | $\tan(\theta)$ | |---|---|---|---| | $0$ | $0$ | $1$ | $0$ | | $\pi/6$ | $1/2$ | $\sqrt{3}/2$ | $1/\sqrt{3}$ | | $\pi/4$ | $\sqrt{2}/2$ | $\sqrt{2}/2$ | $1$ | | $\pi/3$ | $\sqrt{3}/2$ | $1/2$ | $\sqrt{3}$ | | $\pi/2$ | $1$ | $0$ | undefined |

Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$.

Cofunction identities: $\sin(\pi/2 - \theta) = \cos(\theta)$; $\cos(\pi/2 - \theta) = \sin(\theta)$.

Sinusoidal Function Template

$f(t) = a\sin(b(t-c)) + d$

| Parameter | Meaning | How to find from context | |---|---|---| | $a$ | Amplitude (vertical stretch) | $a = \frac{\text{max} - \text{min}}{2}$ | | $b$ | Frequency (period = $2\pi / b$) | Period $T$ → $b = 2\pi / T$ | | $c$ | Phase shift (horizontal, to the right) | When does max or min occur? Set $b(t-c) = \pi/2$ for max | | $d$ | Midline (vertical shift) | $d = \frac{\text{max} + \text{min}}{2}$ |

⚠️ CRITICAL TRAP: The phase shift is $c$, and it shifts the input to the right. Do NOT write $a\sin(bt - c) + d$ — always keep the form $a\sin(b(t-c)) + d$.

Inverse Trigonometric Functions

| Function | Domain | Range | |---|---|---| | $\sin^{-1}(x)$ | $[-1, 1]$ | $[-\pi/2, \pi/2]$ | | $\cos^{-1}(x)$ | $[-1, 1]$ | $[0, \pi]$ | | $\tan^{-1}(x)$ | $\mathbb{R}$ | $(-\pi/2, \pi/2)$ |

Do not confuse $\sin^{-1}(x)$ with $\frac{1}{\sin(x)} = \csc(x)$.

Parametric & Polar Conversions

| Type | Formula | |---|---| | Parametric → Cartesian | Eliminate $t$: if $x = t^2$ and $y = 2t+1$, solve for $t = (y-1)/2$, then $x = ((y-1)/2)^2 = (y-1)^2/4$ | | Polar → Cartesian | $x = r\cos(\theta)$; $y = r\sin(\theta)$; $r^2 = x^2 + y^2$; $\tan(\theta) = y/x$ | | Cartesian → Polar | From $x^2 + y^2 = r^2$ and $x = r\cos(\theta)$ | | Example: $r = 4\cos(\theta)$ | Multiply both sides by $r$: $r^2 = 4r\cos(\theta) \Rightarrow x^2 + y^2 = 4x \Rightarrow (x-2)^2 + y^2 = 4$ (circle) |

Matrix Operations (Quickest Review)

2×2 matrix inverse: $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

Solve $A{\bf x} = {\bf b}$: ${\bf x} = A^{-1}{\bf b}$ (if $A$ is invertible).

Determinant: $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$.

Top 10 Traps That Cost Real Points

1. Phase shift sign error: $(x - c)$ shifts right. Writing $\sin(bx - c)$ is wrong form.
2. Inverse trig vs reciprocal: $\sin^{-1}(x) \neq 1/\sin(x)$.
3. Domain restrictions: $\ln(x)$ only for $x > 0$; $\sin^{-1}(x)$ only for $-1 \leq x \leq 1$.
4. Log properties apply only to positive arguments: $\ln(x^2) = 2\ln|x|$, not $2\ln(x)$ if $x < 0$.
5. Asymptotes vs holes: A hole is removable (common factor); an asymptote is non-removable.
6. Forgetting the initial parameter: In $A(t) = a \cdot b^t$, don't skip finding $a$ (initial value).
7. Period vs frequency confusion: If period is 12, then $b = 2\pi/12 = \pi/6$, not $b = 12$.
8. Parametric domain: $t \in [0, 2\pi]$ typically traces a complete curve; check your domain.
9. Polar angle quadrants: When converting $r = -3$, the point is in the opposite direction (flip 180°).
10. Forgetting to simplify matrix inverse: Always check that $ad - bc \neq 0$ before inverting.

Calculator Must-Do Practice (Tonight)

1. Graph a sinusoidal: Plot $y = 3\sin(2(x-\pi/4)) - 1$ and verify amplitude, period, phase shift visually.
2. Solve numerically: Use solver to find $2^x = 50$ to 3 decimal places.
3. Matrix operations: Enter a 2×2 matrix and compute its inverse using calc menus.
4. Intersections: Graph $y = 2\sin(x)$ and $y = 0.5x$ and find intersections with calc.

Battery check: Replace batteries NOW if your calc has been used heavily. Bring spares.

Score Boundaries (Recent Years)

Approximate score ranges out of 120 total points:

• 5: ~84+
• 4: ~70–83
• 3: ~56–69
• 2: ~42–55
• 1: below ~42

You only need ~70% to score a 4, and ~84% for a 5. You can miss several questions and still earn a top score. Don't panic.

Morning-of Checklist

• ☐ 8–9 hours of sleep.
• ☐ Real breakfast (protein + slow carbs, not just sugar).
• ☐ 2 sharpened #2 pencils, blue/black pen.
• ☐ Graphing calculator + fresh batteries + backup batteries.
• ☐ Photo ID + AP registration label.
• ☐ Water bottle, snack for break.
• ☐ Arrive 30 minutes early.

Exam Day Strategy

Multiple Choice (55 min, no calculator for first 25 min):

• Mark and skip anything taking >90 seconds. Return to it.
• Use process of elimination aggressively; wrong answers often have sign errors.

Multiple Choice (50 min, calculator allowed):

• Use graphs and solver to verify algebraic work.
• Check at least one boundary value in real-world contexts.

Free Response (90 min for 4 FRQs):

• Read all 4 FRQs first. Start with the one you're most confident on.
• Write the expression before plugging in numbers (e.g., write the integral, then evaluate).
• Always define variables: "Let $t$ = years since 2020."
• Never leave a question blank. Partial credit adds up — even setting up the right equation earns points.
• If stuck, move on; you can return in the last 10 minutes.

One Last Thing

You've prepared. The knowledge is there. Trust your work. Show up rested, read carefully, and remember: the rubric wants to give you points—your job is to write clearly enough that it can.

You've got this. 🎯

Need more unit review? Browse the AP Precalculus topic library → or return to FRQ practice guide →.