Comprehensive College Algebra Formula Sheet: Central Texas College

College Algebra Formula Sheet: Central Texas College

Overview

• College: Central Texas College
• Compiler: Amanda Russell
• Scope: Comprehensive formula reference for College Algebra including foundational algebra, functions, polynomials, exponential/logarithmic functions, sequences, probability, trigonometry, and analytic geometry.

Chapter P/1: Foundational Algebra and Quadratic Equations

• Quadratic Formula: Used to determine the solutions for any quadratic equation of the form $ax^2 + bx + c = 0$.
      * $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Exponent Rules:
      * Product Rule: When multiplying like bases, add the exponents.
          * $a^m \cdot a^n = a^{m+n}$
      * Quotient Rule: When dividing like bases, subtract the exponents.
          * $\frac{a^m}{a^n} = a^{m-n}$
      * Power Rule: When raising a power to another power, multiply the exponents.
          * $(a^m)^n = a^{mn}$
      * Negative Exponent Rule: A base with a negative exponent is equivalent to the reciprocal of the base with a positive exponent.
          * $a^{-n} = \frac{1}{a^n}$

Chapter 2: Functions, Lines, and Transformations

• Composite Functions: The composition of a function $f$ with function $g$ is defined as applying function $g$ first and then function $f$.
      * $(f \circ g)(x) = f(g(x))$
• Slope and Line Forms:
      * Slope Formula: Measures the steepness or gradient between two points $(x_1, y_1)$ and $(x_2, y_2)$.
          * $m = \frac{y_2 - y_1}{x_2 - x_1}$
      * Point-Slope Form: Useful for finding the equation of a line passing through $(x_1, y_1)$ with a known slope $m$.
          * $y - y_1 = m(x - x_1)$
      * Slope-Intercept Form: The standard linear equation showing the slope $m$ and y-intercept $b$.
          * $y = mx + b$
• Function Transformations:
      * Vertical Shifts: Moving the graph up or down.
          * $f(x) \pm c$
      * Horizontal Shifts: Moving the graph left or right.
          * $f(x \pm c)$
      * Reflections:
          * Across the x-axis: $-f(x)$
          * Across the y-axis: $f(-x)$
      * Vertical Adjustments:
          * Vertical Stretch: Occurs when $a \cdot f(x)$ where a > 1.
          * Vertical Shrink: Occurs when $a \cdot f(x)$ where $0 < a < 1$.     * Horizontal Adjustments:         * Horizontal Stretch: Occurs when $f(a \cdot x)$ where $0 < a < 1$.         * Horizontal Shrink: Occurs when $f(a \cdot x)$ where $a > 1$.
• Symmetry:
      * Even Symmetry: Symmetry with respect to the y-axis.
          * $f(-x) = f(x)$
      * Odd Symmetry: Symmetry with respect to the origin.
          * $f(-x) = -f(x)$
• Inverse Functions:
      * Definition: $y = f^{-1}(x)$
      * Procedure: Switch the variables $x$ and $y$, then solve for $y$.

Chapter 3: Polynomial and Rational Functions

• Polynomial Zeros:
      * The factor $(x - a)^k$ represents a repeated zero at $x = a$.
      * Odd Multiplicity ($k$): The graph crosses the x-axis at $(a, 0)$.
      * Even Multiplicity ($k$): The graph touches the x-axis (tangent) at $(a, 0)$.
• Rational Asymptotes: For a rational function defined as $f(x) = \frac{N(x)}{D(x)}$:
      * Vertical Asymptotes: These occur at the zeros of the denominator function $D(x)$.
      * Horizontal Asymptotes:
          * If the degree of the numerator is less than the degree of the denominator ($deg(N) < deg(D)$), the asymptote is $y = 0$.         * If the degree of the numerator equals the degree of the denominator ($deg(N) = deg(D)$), the asymptote is $y = \frac{a_n}{b_n}$.         * If the degree of the numerator is greater than the degree of the denominator ($deg(N) > deg(D)$), there is no horizontal asymptote.

Chapter 4: Exponential and Logarithmic Functions and Interest

• Compound Interest:
      * For $n$ compounding periods per year: $A = P \left(1 + \frac{r}{n}\right)^{nt}$
      * For continuous compounding: $A = Pe^{rt}$
      * Variables: $t = \text{time}$, $r = \text{rate}$.
• Exponential Growth and Decay:
      * General Growth Model: $A = A_0 e^{kt}$
      * Half-life Equation: $\ln\left(\frac{1}{2}\right) = kt$
• Logarithm Rules:
      * Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
      * Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
      * Power Rule: $\log_b(x^r) = r \log_b(x)$
      * Change of Base Formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

Chapter 11: Sequences and Series

• Arithmetic Sequences:
      * General Term: $a_n = a_1 + (n - 1)d$
      * Sum of first $n$ terms: $S_n = \frac{n}{2} (a_1 + a_n)$
• Geometric Sequences:
      * General Term: $a_n = a_1 r^{n-1}$
      * Finite Sum ($n$ terms): $S_n = \frac{a_1(1 - r^n)}{1 - r}$
      * Infinite Geometric Sum: $S = \frac{a_1}{1 - r}$

Principles of Probability and Counting

• Types of Probability:
      * Classical Probability: P(E) = \frac{# \text{ outcomes in } E}{\text{total outcomes in sample space}}
      * Empirical/Statistical Probability: $P(E) = \frac{\text{Frequency of event } E}{\text{Total frequency}} = \frac{f}{n}$
• Probability Rules:
      * Complement Rule: $P(E') = 1 - P(E)$
      * Probability of Both Events (A and B):
          * General case: $P(A \cap B) = P(A) \cdot P(B | A)$
          * Independent events: $P(A \cap B) = P(A) \cdot P(B)$
      * Probability of Either A or B or Both (Union):
          * General case: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
          * Mutually exclusive events: $P(A \cup B) = P(A) + P(B)$
• Counting Principles:
      * Permutations: Arrangement of $r$ items from $n$ where order matters ($r \le n$).
          * $_n P_r = \frac{n!}{(n - r)!}$
      * Combinations: Selection of $r$ items from $n$ where order does not matter ($r \le n$).
          * $_n C_r = \frac{n!}{(n - r)!r!}$

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Trigonometric Functions and Identities

• The Six Trigonometric Functions:
      * $\sin(\theta) = \frac{\text{opp}}{\text{hyp}} = \frac{y}{r} = \frac{1}{\csc(\theta)}$
      * $\cos(\theta) = \frac{\text{adj}}{\text{hyp}} = \frac{x}{r} = \frac{1}{\sec(\theta)}$
      * $\tan(\theta) = \frac{\text{opp}}{\text{adj}} = \frac{y}{x} = \frac{1}{\cot(\theta)} = \frac{\sin(\theta)}{\cos(\theta)}$
      * $\csc(\theta) = \frac{\text{hyp}}{\text{opp}} = \frac{r}{y} = \frac{1}{\sin(\theta)}$
      * $\sec(\theta) = \frac{\text{hyp}}{\text{adj}} = \frac{r}{x} = \frac{1}{\cos(\theta)}$
      * $\cot(\theta) = \frac{\text{adj}}{\text{opp}} = \frac{x}{y} = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$
• Circular Geometry:
      * Arc Length: $s = r \theta$
      * Area of a Sector: $A = \frac{1}{2} \pi r^2 \theta$

Analytic Geometry: Conic Sections

• Parabola:
      * Vertical Orientation: $(x - h)^2 = 4p(y - k)$ with directrix $y = k - p$
      * Horizontal Orientation: $(y - k)^2 = 4p(x - h)$ with directrix $x = h - p$
      * Vertex at the Origin: $x^2 = 4py$ or $y^2 = 4px$
• Ellipse:
      * Horizontal Major Axis: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$
      * Vertical Major Axis: $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$
      * Note: In an ellipse, a > b.
• Hyperbola:
      * Horizontal Orientation: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$
      * Vertical Orientation: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$
      * Coordinate Relation: $c^2 = a^2 + b^2$