AP Precalculus Last-Minute Review (Night Before)

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

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

Calculator Must-Do Practice (Tonight)

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