Algebra 2 Cheatsheet

Cheatsheet Content

### Solving Equations & Inequalities - **Linear Equations:** $ax + b = c \implies ax = c - b \implies x = \frac{c-b}{a}$ - **Absolute Value Equations:** $|ax + b| = c \implies ax + b = c$ or $ax + b = -c$ - Always check for extraneous solutions. - **Absolute Value Inequalities:** - $|ax + b| c \implies ax + b > c$ or $ax + b 0$: Two distinct real solutions - $D = 0$: One real solution (repeated) - $D ### Functions - **Definition:** A relation where each input (x) has exactly one output (y). - **Domain:** Set of all possible input values. - **Range:** Set of all possible output values. - **Vertical Line Test:** If any vertical line intersects the graph more than once, it's not a function. - **Function Notation:** $y = f(x)$ - **Inverse Functions:** $f^{-1}(x)$ - To find: swap $x$ and $y$, then solve for $y$. - Property: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ - Graphs are reflections over the line $y=x$. - **Function Operations:** - $(f+g)(x) = f(x) + g(x)$ - $(f-g)(x) = f(x) - g(x)$ - $(f \cdot g)(x) = f(x) \cdot g(x)$ - $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}, g(x) \neq 0$ - **Composition of Functions:** $(f \circ g)(x) = f(g(x))$ ### Polynomials - **Degree:** Highest exponent of the variable. - **Leading Coefficient:** Coefficient of the term with the highest degree. - **End Behavior:** Determined by the degree and leading coefficient. - Even degree, positive leading coefficient: up-up - Even degree, negative leading coefficient: down-down - Odd degree, positive leading coefficient: down-up - Odd degree, negative leading coefficient: up-down - **Rational Root Theorem:** Possible rational roots $p/q$, where $p$ divides the constant term and $q$ divides the leading coefficient. - **Remainder Theorem:** If a polynomial $P(x)$ is divided by $(x-c)$, the remainder is $P(c)$. - **Factor Theorem:** $(x-c)$ is a factor of $P(x)$ if and only if $P(c) = 0$. - **Fundamental Theorem of Algebra:** A polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicity). ### Exponents & Logarithms - **Exponent Rules:** - $a^m \cdot a^n = a^{m+n}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $(a^m)^n = a^{mn}$ - $(ab)^n = a^n b^n$ - $a^0 = 1$ ($a \neq 0$) - $a^{-n} = \frac{1}{a^n}$ - $(\frac{a}{b})^n = \frac{a^n}{b^n}$ - **Logarithm Definition:** $\log_b y = x \iff b^x = y$ - **Common Log:** $\log x = \log_{10} x$ - **Natural Log:** $\ln x = \log_e x$ - **Logarithm Rules:** - $\log_b (MN) = \log_b M + \log_b N$ - $\log_b (\frac{M}{N}) = \log_b M - \log_b N$ - $\log_b (M^p) = p \log_b M$ - $\log_b b = 1$ - $\log_b 1 = 0$ - $\log_b b^x = x$ - $b^{\log_b x} = x$ - **Change of Base Formula:** $\log_b M = \frac{\log_a M}{\log_a b}$ ### Rational Expressions & Equations - **Simplifying:** Factor numerator and denominator, then cancel common factors. - **Multiplying:** $\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}$ - **Dividing:** $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{AD}{BC}$ - **Adding/Subtracting:** Find a Common Denominator (LCD), then add/subtract numerators. - **Solving Rational Equations:** 1. Find LCD of all terms. 2. Multiply every term by the LCD to clear denominators. 3. Solve the resulting equation. 4. **Crucially, check for extraneous solutions** (values that make original denominators zero). ### Sequences & Series - **Arithmetic Sequence:** $a_n = a_1 + (n-1)d$ - $a_1$: first term, $d$: common difference - **Sum of first $n$ terms:** $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(2a_1 + (n-1)d)$ - **Geometric Sequence:** $a_n = a_1 \cdot r^{n-1}$ - $r$: common ratio - **Sum of first $n$ terms:** $S_n = a_1 \frac{1-r^n}{1-r}$, $r \neq 1$ - **Sum of infinite geometric series:** $S = \frac{a_1}{1-r}$, $|r|