Before You Start
📚 Core Concepts & Key Formulas
Unit 1 · Functions
Functions, Domain & Range
A function assigns exactly one output to each input. The domain is the set of all valid inputs; the range is all possible outputs.
Vertical Line Test: a graph is a function iff no vertical line crosses it more than once
🔑 Memorize: For \(\sqrt{f(x)}\), set \(f(x)\geq 0\). For \(\frac{1}{f(x)}\), set \(f(x)\neq 0\).
Quick Example
Find the domain of \(f(x)=\sqrt{x-3}\).
✔ Set \(x-3\geq 0\Rightarrow x\geq 3\). Domain: \([3,\infty)\)
Unit 2 · Transformations
Transformations of Functions
\(y=af(b(x-h))+k\)
🔑 Memorize: \(+h\) → shift right, \(+k\) → shift up, \(|a|>1\) → stretch, \(0<|a|<1\) → compress, \(a<0\) → reflect over x-axis, \(b<0\) → reflect over y-axis.
Quick Example
\(g(x)=-2f(x+3)-1\): reflect, vertical stretch ×2, shift left 3, down 1.
✔ Vertex shifts from \((0,0)\) to \((-3,-1)\)
Unit 3 · Polynomials
Polynomial Functions & Zeros
Factor Theorem: \((x-c)\) is a factor \(\Leftrightarrow f(c)=0\)
🔑 Memorize: Degree \(n\) → at most \(n\) real zeros. Odd degree → crosses x-axis; even degree → touches and turns at repeated zeros.
Quick Example
Find zeros of \(f(x)=x^3-x^2-6x\).
✔ \(f(x)=x(x-3)(x+2)\) → zeros: \(x=0,3,-2\)
Unit 4 · Rational Functions
Rational Functions & Asymptotes
HA: compare degrees of numerator \((m)\) and denominator \((n)\)
🔑 Memorize: \(m<n\): HA at \(y=0\). \(m=n\): HA at \(y=\frac{\text{leading coeff}_{\text{num}}}{\text{leading coeff}_{\text{den}}}\). \(m>n\): no HA. VA: set denominator \(=0\).
Quick Example
\(f(x)=\frac{3x^2}{x^2-4}\): HA = \(y=3\), VA: \(x=\pm2\)
✔ Same degree → HA = \(\frac{3}{1}=3\)
Unit 5 · Exp & Log
Exponential & Logarithmic Functions
\(\log_b(xy)=\log_b x+\log_b y\quad\log_b\!\left(\tfrac{x}{y}\right)=\log_b x-\log_b y\quad\log_b(x^r)=r\log_b x\)
🔑 Memorize: \(b^x=y \Leftrightarrow \log_b y=x\). Change of base: \(\log_b a=\frac{\ln a}{\ln b}\). \(e^{\ln x}=x\).
Quick Example
Solve \(2^{x+1}=32\).
✔ \(2^{x+1}=2^5 \Rightarrow x+1=5 \Rightarrow x=4\)
Unit 6 · Trigonometry
Trigonometric Functions & Identities
\(\sin^2\theta+\cos^2\theta=1\quad\tan\theta=\frac{\sin\theta}{\cos\theta}\)
\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\quad\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\)
🔑 Memorize (Unit Circle): 0°(1,0), 30°(\(\frac{\sqrt3}{2},\frac12\)), 45°(\(\frac{\sqrt2}{2},\frac{\sqrt2}{2}\)), 60°(\(\frac12,\frac{\sqrt3}{2}\)), 90°(0,1)
Quick Example
\(\sin(30°)=\frac{1}{2},\ \cos(60°)=\frac{1}{2},\ \tan(45°)=1\)
Unit 7 · Trig Graphs
Amplitude, Period & Phase Shift
\(y=A\sin(Bx-C)+D\quad\text{Period}=\frac{2\pi}{|B|}\quad\text{Amplitude}=|A|\quad\text{Phase shift}=\frac{C}{B}\)
🔑 Memorize: Amplitude = half the vertical distance from max to min. Period = length of one full cycle. For \(\tan\): period = \(\frac{\pi}{|B|}\).
Quick Example
\(y=3\sin(2x-\pi)+1\): Amplitude=3, Period=\(\pi\), Phase shift=\(\frac{\pi}{2}\) right, Vertical shift=1 up.
Unit 8 · Inverse Trig
Inverse Trigonometric Functions
\(\arcsin:\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\quad\arccos:[0,\pi]\quad\arctan:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)
🔑 Memorize: \(\sin^{-1}(\frac{\sqrt{3}}{2})=\frac{\pi}{3}\), \(\cos^{-1}(\frac{1}{2})=\frac{\pi}{3}\), \(\tan^{-1}(1)=\frac{\pi}{4}\)
Quick Example
Evaluate \(\arctan(\sqrt{3})\).
✔ \(\tan\frac{\pi}{3}=\sqrt{3}\Rightarrow\arctan(\sqrt{3})=\frac{\pi}{3}\)
Unit 9 · Conics
Conic Sections
Circle: \((x-h)^2+(y-k)^2=r^2\) Parabola: \(y=a(x-h)^2+k\)
Ellipse: \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\) Hyperbola: \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\)
🔑 Memorize: For ellipse: \(a>b>0\), major axis along \(x\) if \(a^2\) is under \(x\). \(c^2=a^2-b^2\) (ellipse), \(c^2=a^2+b^2\) (hyperbola).
Quick Example
Center and radius of \((x-2)^2+(y+3)^2=25\)?
✔ Center \((2,-3)\), radius \(=5\)
Unit 10 · Sequences
Arithmetic & Geometric Sequences
Arithmetic: \(a_n=a_1+(n-1)d\quad S_n=\frac{n}{2}(a_1+a_n)\)
Geometric: \(a_n=a_1\cdot r^{n-1}\quad S_n=\frac{a_1(1-r^n)}{1-r}\ (r\neq1)\quad S_\infty=\frac{a_1}{1-r}\ (|r|<1)\)
🔑 Memorize: Common difference \(d=a_{n+1}-a_n\). Common ratio \(r=\frac{a_{n+1}}{a_n}\). Infinite geometric sum exists only when \(|r|<1\).
Quick Example
Sum of infinite geometric series with \(a_1=6,\ r=\frac{1}{2}\).
✔ \(S_\infty=\frac{6}{1-\frac{1}{2}}=12\)