Algebra 1 EOC Comprehensive Cheat Sheet

Equations: When solving equations, think about opposites (inverse operations).
Fractions:
- When dealing with multiple fractions, you can multiply by the common denominator to cancel them out.
- To cancel a single fraction, multiply by the reciprocal, which is the flipped version of the fraction.
Proportions: To solve a proportion where one fraction equals another fraction, cross-multiply.
Inequalities: When solving inequalities, remember to flip the inequality sign around if you multiply or divide both sides by a negative number.

Definition: A function is a relation where each input value has exactly one output value.
Vertical Line Test: This test can be used with graphs to determine if a relation is a function or not.
Domain:
- Definitions: The set of $x$ values, input values, or the independent variable.
- Format for Discrete Domain: $\{ 1, 2, 3, 4 \}$ (set of specific points).
- Format for Continuous Domain: $-4 \le x \le 5$ (indicating $x$ is between $-4$ and $5$ ).
Range:
- Definitions: The set of $y$ values, output values, or the dependent variable.
- Format for Discrete Range: $\{ 1, 2, 3, 4 \}$ (set of specific points).
- Format for Continuous Range: $0 \le y \le 10$ (indicating $y$ is between $0$ and $10$ ).
Continuous Relation: A relation that would have no "breaks" if it were to be graphed.
Discrete Relation: A relation that would have breaks in between the set points if it were to be graphed.

Rate of Change: This is another term for slope.
Vertical Lines:
- Format of Equation: $x = \text{\_\_\_}$
- Slope: Undefined.
Horizontal Lines:
- Format of Equation: $y = \text{\_\_\_}$
- Slope: 0.
Parallel Slopes: Parallel lines have the same slope.
Perpendicular Slopes: Perpendicular lines have opposite reciprocal slopes. (Example: Slopes of $\frac{2}{3}$ and $-\frac{3}{2}$ are perpendicular).

Intercepts:
- To find an $x$ -intercept, plug in $0$ for $y$ .
- To find a $y$ -intercept, plug in $0$ for $x$ .
Linear Equation Formats:
- Point-Slope-Form: $y - y_1 = m(x - x_1)$
- Slope-Intercept Form: $y = mx + b$
- Standard Form: $Ax + By = C$
  - Procedures: Get $x$ and $y$ together on one side of the equal sign.
  - Constraint 1: $A$ , $B$ , and $C$ cannot be left as fractions.
  - Constraint 2: $A$ must be positive. If $A$ is negative, flip all signs in the equation.

Translations:
- $f(x) + a$ : The graph moves up "a" units.
- $f(x) - a$ : The graph moves down "a" units.
- $f(x + a)$ : The graph moves left "a" units.
- $f(x - a)$ : The graph moves right "a" units.
Stretch and Compression:
- Vertical ( $a \cdot f(x)$ ): Stretch or compress vertically by a factor of "a".
  - If $a > 1$ , it is a stretch.
  - If $a < 1$ , it is a compression.
- Horizontal ( $f(bx)$ ): Stretch or compress horizontally (functions as the opposite of what is intuitive).
  - If $b > 1$ , it is a compression.
  - If $b < 1$ , it is a stretch.
Transformations Specific to Linear Equations ( $y = mx + b$ ):
- $a \cdot f(x)$ : Scales (multiplies) both the slope and the $y$ -intercept by "a".
- $f(bx)$ : Scales only the slope by "b".
Reflections:
- $-f(x)$ : Reflect across the $x$ -axis (Reflect Vertically).
- $f(-x)$ : Reflect across the $y$ -axis (Reflect Horizontally).

Arithmetic Sequences: Each consecutive number has a common difference ( $d$ ), meaning you add or subtract the same value each time.
- Recursive Form: $A_n = a_{n-1} + d$ (Explains what to do to the previous term to get the next term).
- Explicit Form: $A_n = a_1 + d(n - 1)$ (Explains what to do with the first term to get the nth term).
Geometric Sequences: Each consecutive number has a common ratio ( $r$ ), meaning you multiply or divide the same value each time.
- Recursive Form: $A_n = a_{n-1} \times d$ (Note: Common ratio logic applies).
- Explicit Form: $A_n = a_1 \cdot r^{n-1}$
Sequence Variable Key:
- $A_n$ : The nth term.
- $a_{n-1}$ : The previous term.
- $a_1$ : The first term.
- $d$ : The common difference.
- $r$ : The common ratio.

Mechanisms to solve a system of 2 equations:
- Substitution.
- Elimination.
- Graphing (Looking for the point of intersection).
Factoring Techniques:
- Always look for a Greatest Common Factor (GCF) first, which may include common variables or coefficients divisible by the same number.
- 2 terms: Check for the difference of squares.
- 3 terms: Standard factoring (Check what multiplies to $a \cdot c$ and adds to the $b$ value).
- 4 terms: Grouping.

Mechanisms to solve quadratics:
- Graphing: Set the quadratic equal to $0$ and look for $x$ -intercepts.
- Factoring: Set the quadratic equal to $0$ and then set each individual factor to $0$ .
- Quadratic Formula: Set the quadratic equal to $0$ and apply the formula.
- Complete the Square: Move the constant to the other side and complete the square using the term $(\frac{b}{2})^2$ .
Quadratic Equation Formats:
- Standard Form: $y = ax^2 + bx + c$ , where $c$ represents the $y$ -intercept.
- Vertex Form: $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex.
- Axis of Symmetry: A vertical line in the format $x = \text{\_\_\_}$ that goes through the vertex, calculated as $x = \frac{-b}{2a}$ .

Equation Format: $y = a \cdot b^x$
Key Variables:
- $a$ : The starting amount.
- $b$ : The growth or decay factor (based on repeated multiplication or division).
Asymptote: The line where an exponential graph "levels out" at.

Factoring Formulas:
- Perfect Square Trinomials: $a^2 + 2ab + b^2 = (a + b)^2$
- Perfect Square Trinomials: $a^2 - 2ab + b^2 = (a - b)^2$
- Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$
Properties of Exponents:
- Product of powers: $a^m \cdot a^n = a^{m+n}$
- Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$
- Power of a power: $(a^m)^n = a^{mn}$
- Rational exponent: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$
- Negative exponent: $a^{-n} = \frac{1}{a^n}$
Linear and Quadratic References:
- Slope of a line ( $m$ ): $m = \frac{y_2 - y_1}{x_2 - x_1}$
- Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
- Axis of Symmetry: $x = \frac{-b}{2a}$