Formula Reference Card

Left to right within same precedence level.

Definition:

Properties:

Distance on Number Line:

Laws of Exponents:

Definition:

Properties:

Principal Root:

Rationalizing Denominators:

Standard Form:

Special Products:

Domain: All values where denominator \(\neq 0\)

Operations:

Complex Fractions:

Standard Form: \(ax + b = 0\)

Solution: \(x = -\frac{b}{a}\)

Strategy: 1. Distribute and combine like terms 2. Move variables to one side, constants to other 3. Divide by coefficient

Percent:

Simple Interest:

Distance-Rate-Time:

Mixture Problems:

Definition:

Standard Form: \(a + bi\) where \(a\) = real part, \(b\) = imaginary part

Powers of \(i\):

Operations:

Complex Conjugate:

Division:

Standard Form: \(ax^2 + bx + c = 0\)

Quadratic Formula:

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

Alternative Forms:

Factored Form:

Vertex Form:

Completing the Square:

Sum and Product of Roots: For \(ax^2 + bx + c = 0\) with roots \(r_1, r_2\):

Rational Equations:

Radical Equations:

Equations with Rational Exponents:

Equations Quadratic in Form:

Properties:

Compound Inequalities:

AND (Intersection):

OR (Union):

Equations:

Inequalities:

Distance Formula:

Midpoint Formula:

Standard Form:

General Form:

Convert General to Standard: Complete the square for both \(x\) and \(y\)

Definition: A relation where each input has exactly one output

Vertical Line Test: If any vertical line intersects graph more than once, it’s NOT a function

Function Notation: \(y = f(x)\) means \(y\) is a function of \(x\)

Domain: All valid input values (look for: division by zero, even roots of negatives, log of non-positive)

Range: All possible output values

Evaluating: \(f(a)\) means substitute \(a\) for every \(x\)

Slope:

Forms of Linear Equations:

Parallel Lines: Same slope \((m_1 = m_2)\)

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

For \(y = f(x)\):

Combined Transformation Order: Horizontal shifts → Horizontal stretch/reflect → Vertical stretch/reflect → Vertical shifts

Increasing: \(f(x_1) < f(x_2)\) when \(x_1 < x_2\)

Decreasing: \(f(x_1) > f(x_2)\) when \(x_1 < x_2\)

Constant: \(f(x_1) = f(x_2)\) for all \(x\) in interval

Even Function: \(f(-x) = f(x)\) (symmetric about y-axis)

Odd Function: \(f(-x) = -f(x)\) (symmetric about origin)

Local Maximum: \(f(c) \geq f(x)\) for all \(x\) near \(c\)

Local Minimum: \(f(c) \leq f(x)\) for all \(x\) near \(c\)

Domain of Composition: All \(x\) in domain of \(g\) where \(g(x)\) is in domain of \(f\)

Difference Quotient:

Standard Form: \(f(x) = ax^2 + bx + c\)

Vertex Form: \(f(x) = a(x - h)^2 + k\)

Vertex:

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

Direction: - \(a > 0\): Opens upward (minimum at vertex) - \(a < 0\): Opens downward (maximum at vertex)

x-intercepts: Solve \(ax^2 + bx + c = 0\)

y-intercept: \((0, c)\)

Standard Form:

Degree: Highest power of \(x\)

Leading Coefficient: \(a_n\)

End Behavior:

Zeros and Multiplicity:

Maximum Turning Points: \(n - 1\) for degree \(n\) polynomial

Division Algorithm:

Where: - \(P(x)\) = dividend - \(D(x)\) = divisor - \(Q(x)\) = quotient - \(R(x)\) = remainder (degree < degree of divisor)

Synthetic Division: For dividing by \((x - c)\):

Remainder Theorem:

Factor Theorem:

Rational Zero Theorem: If \(P(x)\) has integer coefficients and \(\frac{p}{q}\) is a rational zero:

Possible Rational Zeros:

Descartes' Rule of Signs: - Positive real zeros: number of sign changes in \(P(x)\) or less by even number - Negative real zeros: number of sign changes in \(P(-x)\) or less by even number

Fundamental Theorem of Algebra: Every polynomial of degree \(n \geq 1\) has exactly \(n\) zeros (counting multiplicity, including complex)

Conjugate Pairs Theorem: If \(a + bi\) is a zero of a polynomial with real coefficients, then \(a - bi\) is also a zero

Standard Form:

Domain: All real numbers except where \(Q(x) = 0\)

Vertical Asymptotes: \(x = a\) where \(Q(a) = 0\) and \(P(a) \neq 0\)

Horizontal Asymptotes:

Slant (Oblique) Asymptote: When \(\deg(P) = \deg(Q) + 1\): Perform polynomial division, asymptote is the quotient

Holes: Occur where both \(P(x) = 0\) and \(Q(x) = 0\) (common factors)

x-intercepts: Where \(P(x) = 0\) (and \(Q(x) \neq 0\))

y-intercept: \(f(0) = \frac{P(0)}{Q(0)}\) if defined

Method: 1. Move all terms to one side (compare to 0) 2. Factor completely 3. Find critical values (zeros of numerator and denominator) 4. Create sign chart with test points 5. Determine solution based on inequality direction 6. Check endpoints (include if \(\leq\) or \(\geq\), exclude if \(<\) or \(>\)) 7. Always exclude values where denominator = 0

Direct Variation:

Inverse Variation:

Joint Variation:

Combined Variation:

One-to-One Function: Each output corresponds to exactly one input

Horizontal Line Test: If any horizontal line intersects graph more than once, function is NOT one-to-one

Inverse Function: \(f^{-1}\) where \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\)

Finding Inverse: 1. Replace \(f(x)\) with \(y\) 2. Swap \(x\) and \(y\) 3. Solve for \(y\) 4. Replace \(y\) with \(f^{-1}(x)\)

Graphs: \(f\) and \(f^{-1}\) are reflections over \(y = x\)

Domain/Range: Domain of \(f\) = Range of \(f^{-1}\), and vice versa

Definition:

Properties: - Domain: \((-\infty, \infty)\) - Range: \((0, \infty)\) - y-intercept: \((0, 1)\) - Horizontal asymptote: \(y = 0\) - \(b > 1\): increasing (growth) - \(0 < b < 1\): decreasing (decay)

Natural Exponential:

Transformations:

Definition:

Special Logarithms: - Common: \(\log(x) = \log_{10}(x)\) - Natural: \(\ln(x) = \log_e(x)\)

Properties: - Domain: \((0, \infty)\) - Range: \((-\infty, \infty)\) - x-intercept: \((1, 0)\) - Vertical asymptote: \(x = 0\)

Logarithm of 1:

Logarithm of Base:

Inverse Properties:

Expanding Logarithms: Use properties to write as sum/difference of logs

Condensing Logarithms: Combine into single logarithm

Exponential Equations:

Same Base Method:

Logarithm Method:

Logarithmic Equations:

Convert to Exponential:

One-to-One Property:

Always check solutions: Logarithm arguments must be positive

Exponential Growth/Decay:

Half-Life:

Where \(h\) = half-life

Doubling Time:

Compound Interest:

Continuous Compounding:

pH Scale:

Richter Scale (Earthquakes):

Decibels:

Methods: 1. Graphing: Solution is intersection point 2. Substitution: Solve one equation for a variable, substitute 3. Elimination: Add/subtract equations to eliminate a variable

Solution Types:

Standard Form:

Method: Eliminate one variable to get system of two equations, then solve

Proper Fraction: degree of numerator < degree of denominator

Decomposition Rules:

Methods: - Substitution - Elimination (may need to add/subtract equations) - Graphing

May have 0, 1, 2, 3, 4, or more solutions

Graphing: 1. Graph boundary of each inequality 2. Shade appropriate region for each 3. Solution is intersection of all shaded regions

Test Point: If point satisfies inequality, shade that side

Objective Function: \(z = ax + by\) (maximize or minimize)

Constraints: Linear inequalities defining feasible region

Corner Point Theorem: Maximum/minimum occurs at corner point of feasible region

Method: 1. Graph constraints to find feasible region 2. Identify corner points 3. Evaluate objective function at each corner 4. Select maximum or minimum value

Definition: Rectangular array of numbers

Dimensions: \(m \times n\) means \(m\) rows and \(n\) columns

Entry: \(a_{ij}\) is element in row \(i\), column \(j\)

Augmented Matrix: Matrix form of a system \([A|B]\)

Row Echelon Form (REF): - Leading 1 in each row - Leading 1 is to the right of leading 1 above - Rows of zeros at bottom

Reduced Row Echelon Form (RREF): - REF conditions, plus: - Leading 1 is only nonzero entry in its column

Back Substitution: Solve from bottom row up

Addition/Subtraction: (same dimensions required)

Scalar Multiplication:

Matrix Multiplication: (columns of A = rows of B)

\(A_{m \times n} \cdot B_{n \times p} = C_{m \times p}\)

Properties: - \(AB \neq BA\) in general (not commutative) - \(A(BC) = (AB)C\) (associative) - \(A(B + C) = AB + AC\) (distributive)

Definition:

Property: \(AI = IA = A\)

Definition: \(A^{-1}\) such that \(AA^{-1} = A^{-1}A = I\)

2×2 Inverse:

Finding Inverse (Gauss-Jordan): 1. Form augmented matrix \([A|I]\) 2. Row reduce to \([I|A^{-1}]\)

Solving Systems: \(AX = B \implies X = A^{-1}B\)

2×2 Determinant:

3×3 Determinant (Expansion by first row):

Properties: - \(\det(AB) = \det(A) \cdot \det(B)\) - \(\det(A^{-1}) = \frac{1}{\det(A)}\) - \(\det(kA) = k^n \det(A)\) for \(n \times n\) matrix - \(A\) is invertible \(\iff \det(A) \neq 0\)

Cramer’s Rule: For \(Ax = b\):

Where \(A_i\) is \(A\) with column \(i\) replaced by \(b\)

Definition: Set of points equidistant from focus and directrix

Vertical Axis (opens up/down):

Horizontal Axis (opens left/right):

Latus Rectum: Length = \(|4p|\)

Definition: Set of points where sum of distances to two foci is constant

Horizontal Major Axis:

Vertical Major Axis:

Eccentricity:

Closer to 0 = more circular; closer to 1 = more elongated

Definition: Set of points where difference of distances to two foci is constant

Horizontal Transverse Axis:

Vertical Transverse Axis:

Eccentricity:

Conjugate Axis: Length = \(2b\)

Transverse Axis: Length = \(2a\)

Discriminant: \(B^2 - 4AC\)

Definition: Ordered list of numbers \(a_1, a_2, a_3, \ldots\)

Explicit Formula: \(a_n = f(n)\)

Recursive Formula: \(a_n\) defined in terms of previous terms

Factorial:

Common Difference: \(d = a_{n+1} - a_n\)

Explicit Formula:

Alternative Form:

Sum of First n Terms:

Common Ratio: \(r = \frac{a_{n+1}}{a_n}\)

Explicit Formula:

Alternative Form:

Sum of First n Terms:

Sum of Infinite Geometric Series (|r| < 1):

Sigma Notation:

Properties:

Useful Formulas:

Principle of Mathematical Induction:

Binomial Coefficient:

Properties:

Pascal’s Triangle Identity:

Binomial Theorem:

General Term (r+1)th term:

Special Cases:

Fundamental Counting Principle: If event A can occur \(m\) ways and event B can occur \(n\) ways, then A and B together can occur \(m \cdot n\) ways.

Permutations (order matters):

Permutations of n objects:

Permutations with repetition:

Where \(n_1 + n_2 + \cdots + n_k = n\)

Combinations (order doesn’t matter):

Relationship:

Definition:

Range:

Complement:

Addition Rule:

Mutually Exclusive Events: \(P(A \cap B) = 0\)

Multiplication Rule (Independent Events):

Conditional Probability:

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