AP Precalculus Cheat Sheet

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Unit 1 & 2: Fundamentals, Equations, and Inequalities

Lines and distance

• Slope: $m=\frac{y_2-y_1}{x_2-x_1}$
• Slope-intercept: $y=mx+b$; point-slope: $y-y_1=m(x-x_1)$; standard: $Ax+By=C$ with slope $m=-\frac{A}{B}$.
• Distance: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$; midpoint: $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.
• Parallel lines: equal slopes. Perpendicular lines: $m_1m_2=-1$.

Quadratics

• Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
• Discriminant $b^2-4ac$: positive gives two real roots, zero gives one repeated root, negative gives two complex roots.
• Vieta: $r_1+r_2=-\frac{b}{a}$ and $r_1r_2=\frac{c}{a}$.

Absolute value (for $a>0$)

• $\lvert u\rvert=a \Rightarrow u=a$ or $u=-a$.
• $\lvert u\rvert<a \Rightarrow -a<u<a$ (“less than means between”).
• $\lvert u\rvert>a \Rightarrow u<-a$ or $u>a$ (“greater than means outside”).

Inequality reminders

• Multiplying/dividing by a negative reverses the inequality sign.
• Polynomial/rational inequalities: move to one side, factor, mark zeros, build a sign chart. Even-multiplicity roots do not change sign.
• Never include denominator zeros; never multiply by a denominator of unknown sign.
• Radical/exponential: always state the domain first; check candidates after squaring.

Symmetry tests

• $y$-axis: replace $x$ with $-x$ (even function).
• Origin: replace $x,y$ with $-x,-y$ (odd function).
• $x$-axis: replace $y$ with $-y$.

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Unit 3: Functions

Domain rules

• Denominators cannot be $0$; even roots need a nonnegative radicand; logs need a positive input.

Average rate of change

$$\frac{f(b)-f(a)}{b-a},\qquad \text{difference quotient: } \frac{f(x+h)-f(x)}{h}$$

Transformations (with $c>0$)

• $f(x)+c$ up, $f(x)-c$ down, $f(x+c)$ left, $f(x-c)$ right.
• $-f(x)$ reflects over $x$-axis; $f(-x)$ reflects over $y$-axis.
• Vertex/start form $y=af(x-h)+k$: larger $\lvert a\rvert$ is steeper; $a<0$ reflects vertically.

Composition and inverses

• $(f\circ g)(x)=f(g(x))$; domain needs $x$ in domain of $g$ and $g(x)$ in domain of $f$.
• Inverses satisfy $f(g(x))=x$ and $g(f(x))=x$. To find: swap $x$ and $y$, solve for $y$.
• Domain of $f$ becomes range of $f^{-1}$, and vice versa. A function has an inverse function only if it is one-to-one (passes the horizontal line test, i.e. injective).

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Unit 4 & 13: Polynomial & Rational Functions and Applications to Optimization

Quadratic vertex

• Vertex at $x=-\frac{b}{2a}$; vertex form $f(x)=a(x-h)^2+k$ has vertex $(h,k)$.
• $a>0$ opens up (minimum); $a<0$ opens down (maximum).

Polynomial end behavior

Degree	Leading coefficient	Left end	Right end
even	positive	up	up
even	negative	down	down
odd	positive	down	up
odd	negative	up	down

Zeros and multiplicity

• Odd multiplicity: graph crosses the axis. Even multiplicity: graph touches and bounces.
• Degree $n$ has exactly $n$ complex zeros (with multiplicity) and at most $n-1$ turning points.
• Remainder Theorem: dividing $f(x)$ by $x-m$ leaves remainder $f(m)$.
• Factor Theorem: $x-m$ is a factor exactly when $f(m)=0$.
• Rational Root Theorem: a rational zero $\frac{p}{q}$ in lowest terms has $p\mid a_0$ and $q\mid a_n$.
• Conjugate pairs: real coefficients force $a-bi$ when $a+bi$ is a zero; rational coefficients force $a-\sqrt{b}$ when $a+\sqrt{b}$ is a zero.

Rational functions $R(x)=\frac{f(x)}{g(x)}$

• $x$-intercepts at zeros of $f$ that are still in the domain; $y$-intercept at $R(0)$.
• A canceled factor gives a hole; an uncanceled denominator factor gives a vertical asymptote.
• Horizontal asymptotes by comparing degrees:
  • denominator degree larger: $y=0$.
  • degrees equal: $y=\frac{\text{leading coeff of }f}{\text{leading coeff of }g}$.
  • numerator degree one larger: slant asymptote from division.
  • numerator degree two or more larger: polynomial asymptote.

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Unit 5: Exponential & Logarithmic Functions

Exponential $y=a{b}^x$

• For $y=b^x$ ($b>0$, $b\ne 1$): domain $(-\infty,\infty)$, range $(0,\infty)$, asymptote $y=0$, passes $(0,1)$.
• $b>1$ is growth; $0<b<1$ is decay. The natural base is $e\approx 2.71828$.
• Transformed form $y=a\cdot b^{x-h}+k$ has horizontal asymptote $y=k$.

Logarithm definition

$$\log_b x=y \iff b^y=x,\qquad \log_e x=\ln x$$

• For $y=\log_b x$: domain $(0,\infty)$, range $(-\infty,\infty)$, vertical asymptote $x=0$, passes $(1,0)$. It is the reflection of $b^x$ over $y=x$.

Log rules ($M,N>0$)

Rule	Formula
Product	$\log_b(MN)=\log_b M+\log_b N$
Quotient	$\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N$
Power	$\log_b(M^r)=r\log_b M$
Change of base	$\log_b M=\frac{\log_a M}{\log_a b}$

• There is no sum rule: $\ln(a+b)\ne \ln a+\ln b$.

Solving strategies

• Bases match: set exponents equal. Bases differ: take $\ln$ of both sides and use the power rule.
• Logs present: combine into one log, then rewrite as an exponential. Always impose the domain first.
• A base $0<b<1$ is decreasing, so log inequalities reverse.

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Unit 6 & 7: Trigonometric Functions

Angles, arcs, and motion

• Conversion: $180^\circ=\pi$ radians.
• Arc length $s=r\theta$; sector area $A=\frac12 r^2\theta$ (with $\theta$ in radians).
• Angular speed $\omega=\frac{\theta}{t}$; linear speed $v=\frac{d}{t}=r\omega$.

Right-triangle ratios (SOH-CAH-TOA)

$$\sin\theta=\frac{\text{opp}}{\text{hyp}},\quad \cos\theta=\frac{\text{adj}}{\text{hyp}},\quad \tan\theta=\frac{\text{opp}}{\text{adj}}$$

• Reciprocals: $\csc\theta=\frac{1}{\sin\theta}$, $\sec\theta=\frac{1}{\cos\theta}$, $\cot\theta=\frac{1}{\tan\theta}$.
• Cofunctions: $\sin\theta=\cos\left(\frac{\pi}{2}-\theta\right)$.

Special triangles

• $30$-$60$-$90$: sides $x:x\sqrt3:2x$.
• $45$-$45$-$90$: sides $x:x:x\sqrt2$.

Unit circle: $P=(\cos \theta ,\sin \theta )$

$\theta$	Degrees	$\cos\theta$	$\sin\theta$
$0$	$0^\circ$	$1$	$0$
$\frac{\pi}{6}$	$30^\circ$	$\frac{\sqrt3}{2}$	$\frac12$
$\frac{\pi}{4}$	$45^\circ$	$\frac{\sqrt2}{2}$	$\frac{\sqrt2}{2}$
$\frac{\pi}{3}$	$60^\circ$	$\frac12$	$\frac{\sqrt3}{2}$
$\frac{\pi}{2}$	$90^\circ$	$0$	$1$
$\pi$	$180^\circ$	$-1$	$0$
$\frac{3\pi}{2}$	$270^\circ$	$0$	$-1$

• First-quadrant values repeat with signs set by the quadrant. Quadrant signs: I all $+$; II only $\sin$ $+$; III only $\tan$ $+$; IV only $\cos$ $+$ (“All Students Take Calculus”).
• Reference angle gives the magnitude; the quadrant gives the sign. Coterminal angles differ by $2\pi$ (or $360^\circ$).

Pythagorean identities

$$\sin^2\theta+\cos^2\theta=1$$ $$1+\tan^2\theta=\sec^2\theta,\qquad 1+\cot^2\theta=\csc^2\theta$$

Even/odd and periods

• $\cos(-\theta)=\cos\theta$ (even); $\sin(-\theta)=-\sin\theta$ and $\tan(-\theta)=-\tan\theta$ (odd).
• Period of $\sin$ and $\cos$ is $2\pi$; period of $\tan$ is $\pi$.

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Unit 8 & 9: Graphs and Analytics of Trig Functions

Sinusoidal transformations $y=A\sin (B(x-C))+D$

• Amplitude $\lvert A\rvert$; period $\frac{2\pi}{\lvert B\rvert}$; phase shift $C$; midline $y=D$.
• Range is $[D-\lvert A\rvert,\,D+\lvert A\rvert]$. If written $y=A\sin(Bx-C)+D$, factor to get phase shift $\frac{C}{B}$.

Addition and subtraction formulas

$$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$$ $$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$$ $$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$$

Double-angle formulas

$$\sin(2\theta)=2\sin\theta\cos\theta$$ $$\cos(2\theta)=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$$ $$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$$

Power-reducing and half-angle formulas

$$\cos^2\theta=\frac{1+\cos(2\theta)}{2},\qquad \sin^2\theta=\frac{1-\cos(2\theta)}{2}$$ $$\sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos\theta}{2}},\qquad \cos\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1+\cos\theta}{2}}$$

Inverse trig ranges

Function	Domain	Range
$\sin^{-1}x$	$[-1,1]$	$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$
$\cos^{-1}x$	$[-1,1]$	$[0,\pi]$
$\tan^{-1}x$	$(-\infty,\infty)$	$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

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Unit 10: Additional Topics in Trigonometry (Triangle Laws, Parametric, Polar, and Vectors)

Triangle laws

• Law of Sines: $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$ (use for ASA, AAS, SSA).
• SSA is the ambiguous case: with $h=b\sin A$, you may get $0$, $1$, or $2$ triangles; remember $\sin\theta=\sin(180^\circ-\theta)$.
• Law of Cosines: $a^2=b^2+c^2-2bc\cos A$ (use for SSS, SAS); rearranged $\cos A=\frac{b^2+c^2-a^2}{2bc}$.
• Area: $K=\frac12 ab\sin C$.

Vectors

• From magnitude and direction: $\mathbf v=\langle\lvert\mathbf v\rvert\cos\theta,\ \lvert\mathbf v\rvert\sin\theta\rangle$.
• Magnitude of $\langle a,b\rangle$: $\lvert\mathbf v\rvert=\sqrt{a^2+b^2}$.
• Unit vector: $\mathbf u=\frac{\mathbf v}{\lvert\mathbf v\rvert}$; standard basis $\mathbf i=\langle 1,0\rangle$, $\mathbf j=\langle 0,1\rangle$.
• Dot product: $\mathbf A\cdot\mathbf B=x_1x_2+y_1y_2=\lvert\mathbf A\rvert\lvert\mathbf B\rvert\cos\theta$, so $\cos\theta=\frac{\mathbf A\cdot\mathbf B}{\lvert\mathbf A\rvert\lvert\mathbf B\rvert}$.

Parametric equations

• $x=f(t)$, $y=g(t)$ trace a curve with direction as $t$ increases.
• Eliminate the parameter by solving one equation for $t$ and substituting.

Polar coordinates

• Conversions: $x=r\cos\theta$, $y=r\sin\theta$, $x^2+y^2=r^2$, $\tan\theta=\frac{y}{x}$, $r=\sqrt{x^2+y^2}$.

Complex numbers and De Moivre

• Polar form: $z=r(\cos\theta+i\sin\theta)$ with $r=\sqrt{a^2+b^2}$.
• De Moivre’s Theorem: $z^n=r^n(\cos(n\theta)+i\sin(n\theta))$.

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Conic Sections

Conic	Standard form	Key facts
Circle	$(x-h)^2+(y-k)^2=r^2$	center $(h,k)$, radius $r$, $e=0$
Parabola (vertical)	$(x-h)^2=4p(y-k)$	vertex $(h,k)$, focus $(h,k+p)$, directrix $y=k-p$, $e=1$
Parabola (horizontal)	$(y-k)^2=4p(x-h)$	vertex $(h,k)$, focus $(h+p,k)$, directrix $x=h-p$
Ellipse (horizontal)	$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$	$a>b$, foci $(h\pm c,k)$, $c^2=a^2-b^2$
Ellipse (vertical)	$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$	foci $(h,k\pm c)$, $c^2=a^2-b^2$
Hyperbola (horizontal)	$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$	foci $(h\pm c,k)$, $c^2=a^2+b^2$, asymptotes $y-k=\pm\frac{b}{a}(x-h)$
Hyperbola (vertical)	$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$	foci $(h,k\pm c)$, $c^2=a^2+b^2$, asymptotes $y-k=\pm\frac{a}{b}(x-h)$

• Eccentricity $e=\frac{c}{a}$: circle $e=0$, parabola $e=1$, ellipse $0<e<1$, hyperbola $e>1$.
• For an ellipse, $a>b>0$ and the foci lie along the major axis; for a hyperbola, $c^2=a^2+b^2$.

Conics in polar (focus at the pole)

$$r=\frac{ed}{1\pm e\cos\theta}\qquad\text{or}\qquad r=\frac{ed}{1\pm e\sin\theta}$$

• $e=1$ parabola, $e<1$ ellipse, $e>1$ hyperbola. The $\cos\theta$ form has a vertical directrix; $\sin\theta$ has a horizontal directrix.

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Additional Topics: Sequences and Series

Sequences

• Arithmetic: $a_n=a_1+(n-1)d$.
• Geometric: $a_n=a_1 r^{\,n-1}$.

Series

• Finite arithmetic: $S_n=\frac{n}{2}(a_1+a_n)=\frac{n}{2}\bigl(2a_1+(n-1)d\bigr)$.
• Finite geometric ($r\ne 1$): $S_n=a_1\frac{1-r^n}{1-r}$.
• Infinite geometric ($\lvert r\rvert<1$): $S=\frac{a_1}{1-r}$; diverges when $\lvert r\rvert\ge 1$.

Binomial Theorem

$$(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k,\qquad \binom{n}{k}=\frac{n!}{k!(n-k)!}$$

• Symmetry: $\binom{n}{k}=\binom{n}{n-k}$; the $n$th row of Pascal’s triangle sums to $2^n$.

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Most Common AP Precalculus Mistakes

1. Forgetting domain restrictions before solving radical, rational, or logarithmic equations.
2. Dropping the second solution of a trig equation (or forgetting the period when listing solutions on an interval).
3. Splitting $\ln(a+b)$ as if there were a sum rule for logarithms.
4. Mixing up even and odd multiplicity when sketching polynomial graphs.
5. Forgetting to reverse a log or exponential inequality when the base is between $0$ and $1$.
6. Using $c^2=a^2-b^2$ for a hyperbola (it is $c^2=a^2+b^2$) or the reverse for an ellipse.
7. Putting an angle in degrees into $s=r\theta$, $A=\frac12 r^2\theta$, or $v=r\omega$ instead of radians.
8. Not checking the SSA ambiguous case for a possible second triangle.