Integers

• Positive whole numbers, negative whole numbers, and zero: \(\{..., -3, -2, -1, 0, 1, 2, 3, ...\}\)

• Adding integers: Same signs → add and keep sign; Different signs → subtract and use sign of larger

• Subtracting integers: \(a - b = a + (-b)\)

• Multiplying/Dividing integers: Same signs → positive; Different signs → negative

Rational Numbers

• Numbers that can be expressed as \(\frac{a}{b}\) where \(a, b\) are integers and \(b \neq 0\)

• Adding/Subtracting: \(\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}\)

• Multiplying: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Dividing: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\)

Roots

• Square root: If \(x^2 = a\), then \(x = \pm\sqrt{a}\)

• Cube root: If \(x^3 = a\), then \(x = \sqrt[3]{a}\)

• Product rule: \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\)

• Quotient rule: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)

Irrational Numbers & Pi

• Irrational numbers: Cannot be expressed as fractions (e.g., \(\sqrt{2}, \sqrt{3}, \pi, e\))

• Pi: \(\pi \approx 3.14159...\) (ratio of circumference to diameter)

Basic Percent Formulas

• Percent: \(\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\%\)

• Part: \(\text{Part} = \frac{\text{Percent}}{100} \times \text{Whole}\)

• Decimal conversion: \(x\% = \frac{x}{100}\)

Simple Interest

• \(I = Prt\)

Where: \(I\) = interest, \(P\) = principal, \(r\) = rate (as decimal), \(t\) = time

• Total Amount: \(A = P + I = P(1 + rt)\)

Percent Change

• \(\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%\)

• Percent Increase: Positive result

• Percent Decrease: Negative result

Angles with Parallel Lines and Transversals

• Corresponding angles: Equal

• Alternate interior angles: Equal

• Alternate exterior angles: Equal

• Consecutive interior angles: Supplementary (sum = 180°)

Triangle Angles

• Sum of interior angles: \(A + B + C = 180°\)

• Exterior angle: \(\text{Exterior angle} = \text{Sum of two non-adjacent interior angles}\)

Pythagorean Theorem

• \(a^2 + b^2 = c^2\) (where \(c\) is the hypotenuse)

• Converse: If \(a^2 + b^2 = c^2\), then triangle is right-angled

Midpoint Formula

• Midpoint between \((x_1, y_1)\) and \((x_2, y_2)\):

\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

Distance Formula

• Distance between \((x_1, y_1)\) and \((x_2, y_2)\):

\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Arithmetic Sequences

• General term: \(a_n = a_1 + (n-1)d\)

Where: \(a_1\) = first term, \(d\) = common difference, \(n\) = term number

• Sum of n terms: \(S_n = \frac{n}{2}(a_1 + a_n)\) or \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)

Geometric Sequences

• General term: \(a_n = a_1 \cdot r^{n-1}\)

Where: \(a_1\) = first term, \(r\) = common ratio, \(n\) = term number

• Sum of n terms: \(S_n = a_1\left(\frac{1-r^n}{1-r}\right)\) where \(r \neq 1\)

The Distributive Property

• \(a(b + c) = ab + ac\)

• \(a(b - c) = ab - ac\)

Factoring Formulas

• \(a^2 - b^2 = (a+b)(a-b)\)

• \(a^2 + 2ab + b^2 = (a+b)^2\)

• \(a^2 - 2ab + b^2 = (a-b)^2\)

• \(x^2 + (a+b)x + ab = (x+a)(x+b)\)

• \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)

• \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

Solving Linear Equations

• Standard form: \(ax + b = c\)

• Solution: \(x = \frac{c-b}{a}\) (where \(a \neq 0\))

• Multi-step: Use inverse operations and combine like terms

Special Cases

• Infinitely many solutions: Both sides simplify to same expression

• No solution: Results in false statement (e.g., \(0 = 5\))

Inequality Properties

• If \(a < b\), then \(a + c < b + c\)

• If \(a < b\) and \(c > 0\), then \(ac < bc\)

• If \(a < b\) and \(c < 0\), then \(ac > bc\) (reverse inequality)

Compound Inequalities

• AND (intersection): \(a < x < b\) means \(x > a\) AND \(x < b\)

• OR (union): \(x < a\) OR \(x > b\)

Absolute Value Definition

• \(|a| = a\) if \(a \geq 0\)

• \(|a| = -a\) if \(a < 0\)

• Properties: \(|a| \geq 0\), \(|-a| = |a|\), \(|ab| = |a||b|\)

Absolute Value Equations

• If \(|x| = a\) (where \(a \geq 0\)), then \(x = a\) or \(x = -a\)

Absolute Value Inequalities

• If \(|x| < a\), then \(-a < x < a\)

• If \(|x| > a\), then \(x < -a\) or \(x > a\)

Functions

• Definition: Each input has exactly one output

• Vertical Line Test: Function if vertical line intersects graph at most once

• Function notation: \(f(x)\) represents output when input is \(x\)

Domain and Range

• Domain: Set of all possible input values (x-values)

• Range: Set of all possible output values (y-values)

Inverse Functions

• If \(f(a) = b\), then \(f^{-1}(b) = a\)

• Finding inverse: Switch \(x\) and \(y\), then solve for \(y\)

• \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\)

Direct Proportion

• \(y = kx\) where \(k\) is the constant of proportionality

• Finding k: \(k = \frac{y}{x}\)

• Graph passes through origin (0, 0)

Slope

• \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\)

• Positive slope: Line rises from left to right

• Negative slope: Line falls from left to right

• Zero slope: Horizontal line

• Undefined slope: Vertical line

Forms of Linear Equations

• Slope-intercept form: \(y = mx + b\)

Where \(m\) = slope, \(b\) = y-intercept

• Point-slope form: \(y - y_1 = m(x - x_1)\)

Where \(m\) = slope, \((x_1, y_1)\) = point on line

• Standard form: \(Ax + By = C\)

Where \(A, B, C\) are integers, \(A \geq 0\)

Parallel and Perpendicular Lines

• Parallel lines: Same slope (\(m_1 = m_2\))

• Perpendicular lines: Slopes are negative reciprocals (\(m_1 \cdot m_2 = -1\))

Systems of Equations

• Substitution method: Solve one equation for one variable, substitute into other

• Elimination method: Add or subtract equations to eliminate a variable

• Graphing method: Solution is intersection point of lines

Linear Inequalities

• Graph as line (solid for \(\leq\) or \(\geq\), dashed for \(<\) or \(>\))

• Shade region above line for \(y >\) or \(y \geq\)

• Shade region below line for \(y <\) or \(y \leq\)

Properties of Exponents

• \(a^m \cdot a^n = a^{m+n}\)

• \(\frac{a^m}{a^n} = a^{m-n}\)

• \((a^m)^n = a^{mn}\)

• \((ab)^n = a^n b^n\)

• \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

• \(a^0 = 1\) (where \(a \neq 0\))

• \(a^{-n} = \frac{1}{a^n}\)

Rational Exponents

• \(a^{\frac{1}{n}} = \sqrt[n]{a}\)

• \(a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\)

Format

• \(a \times 10^n\) where \(1 \leq |a| < 10\) and \(n\) is an integer

Operations

• Multiplication: \((a \times 10^m)(b \times 10^n) = (a \cdot b) \times 10^{m+n}\)

• Division: \(\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}\)

• Addition/Subtraction: Convert to same power of 10, then add/subtract

General Form

• \(y = a \cdot b^x\) where \(a \neq 0\), \(b > 0\), \(b \neq 1\)

• \(a\) = initial value, \(b\) = growth/decay factor

Exponential Growth

• \(y = a(1 + r)^t\) where \(r > 0\)

• \(b > 1\) indicates growth

• \(r\) = growth rate (as decimal)

Exponential Decay

• \(y = a(1 - r)^t\) where \(0 < r < 1\)

• \(0 < b < 1\) indicates decay

• \(r\) = decay rate (as decimal)

Standard Form

• \(ax^2 + bx + c = 0\) where \(a \neq 0\)

The Quadratic Formula

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• Discriminant: \(\Delta = b^2 - 4ac\)

- If \(\Delta > 0\): Two real, distinct solutions

- If \(\Delta = 0\): One real, repeated solution

- If \(\Delta < 0\): Two complex solutions

Factoring Quadratics

• \(x^2 + (a+b)x + ab = (x+a)(x+b)\)

• \(ax^2 + bx + c = a(x - r_1)(x - r_2)\) where \(r_1, r_2\) are roots

Completing the Square

• Convert \(ax^2 + bx + c = 0\) to \(a(x - h)^2 + k = 0\)

• Steps: Divide by \(a\), move constant, add \(\left(\frac{b}{2a}\right)^2\) to both sides

Quadratic Functions

• Vertex form: \(y = a(x - h)^2 + k\)

Vertex: \((h, k)\); Opens up if \(a > 0\), down if \(a < 0\)

• Standard form: \(y = ax^2 + bx + c\)

Vertex: \(\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\)

• Axis of symmetry: \(x = h\) or \(x = -\frac{b}{2a}\)

Absolute Value Functions

• Parent function: \(f(x) = |x|\)

• General form: \(f(x) = a|x - h| + k\)

Vertex: \((h, k)\); Opens up if \(a > 0\), down if \(a < 0\)

Piecewise Functions

• Function defined by different formulas on different intervals

• Evaluate by determining which condition the input satisfies

Measures of Center and Spread

• Mean: \(\bar{x} = \frac{\sum x_i}{n}\)

• Median: Middle value when data is ordered

• Mode: Most frequent value

• Range: \(\text{Maximum} - \text{Minimum}\)

Standard Deviation and Variance

• Variance: \(s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\)

• Standard deviation: \(s = \sqrt{s^2} = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}\)

Box and Whisker Plots

• Shows: Minimum, Q1 (25th percentile), Median (Q2), Q3 (75th percentile), Maximum

• IQR (Interquartile Range): \(\text{IQR} = Q_3 - Q_1\)

• Outliers: Values below \(Q_1 - 1.5 \times \text{IQR}\) or above \(Q_3 + 1.5 \times \text{IQR}\)

Scatter Plots and Correlation

• Positive correlation: As \(x\) increases, \(y\) increases

• Negative correlation: As \(x\) increases, \(y\) decreases

• No correlation: No apparent relationship

• Correlation coefficient: \(r\) ranges from -1 to 1

Line of Best Fit

• Line that best represents data on scatter plot

• Also called "regression line" or "trend line"

• Form: \(y = mx + b\) where \(m\) and \(b\) minimize squared residuals