Comprehensive College Mathematics Study Guide: Algebra, Functions, Coordinate Geometry, Trigonometry, and Statistics

Algebraic Expressions, Distribution, and Factoring Identities

• Simplifying and Combining Like Terms:

  • Adding linear variable terms:     $x + 2x + 5x = 8x$

  • Combining algebraic constant and linear terms:     $6x + 4$

  • Multivariable linear combinations:     $-3x + 5 - y$

• Expanding Polynomials via the Distributive Property:

  • Distributing a monomial with a negative coefficient across a multivariate trinomial:     $-2x(5y - 3x + 9)$

  • Multiplying $-2x$ by $5y$ yields $-10xy$.

  • Multiplying $-2x$ by $-3x$ yields $+6x^2$.

  • Multiplying $-2x$ by $+9$ yields $-18x$.

  • Resulting expanded polynomial expression:       $-10xy + 6x^2 - 18x$

  • Multiplying binomial and trinomial polynomials:     $(2x + 9)(5x + y + 4)$

  • Distribute $2x$ through the trinomial:       $2x(5x) + 2x(y) + 2x(4) = 10x^2 + 2xy + 8x$

  • Distribute $9$ through the trinomial:       $9(5x) + 9(y) + 9(4) = 45x + 9y + 36$

  • Sum and group like terms:       $10x^2 + 2xy + (8x + 45x) + 9y + 36 = 10x^2 + 2xy + 53x + 9y + 36$

• Core Algebraic Factoring Formulas:

  • Square of a binomial sum:     $(a + b)^2 = a^2 + 2ab + b^2$

  • Square of a binomial difference:     $(a - b)^2 = a^2 - 2ab + b^2$

  • Difference of two squares:     $a^2 - b^2 = (a - b)(a + b)$

• Step-by-Step Solutions for Linear and Literal Equations:

  • Single-step division with a negative coefficient:     $-1x = 5$     $x = \frac{5}{-1} = -5$

  • Two-step linear equations:     $2x + 5 = 9$     $2x = 9 - 5 = 4$     $x = \frac{4}{2} = 2$

  • Equivalent two-step structure:     $2x + 1 = 5$     $2x = 5 - 1 = 4$     $x = \frac{4}{2} = 2$

  • Linear equation with a negative leading coefficient:     $-2x + 5 = 9$     $-2x = 9 - 5 = 4$     $x = \frac{4}{-2} = -2$

  • Binomial grouped linear equation:     $5(a + b) - 1 = 9$     $5(a + b) = 9 + 1 = 10$     $a + b = \frac{10}{5} = 2$

  • Literal equation solved for variable $x$:     $-ax + b = c$     $-ax = c - b$     $x = \frac{c - b}{-a} = \frac{b - c}{a}$

Linear Inequalities and Verbal Translation Rules

• Mathematical Translation of Verbal Inequality Phrases:

  • "Greater than":     $x > 10$

  • "Smaller than" or "Less than":     $x < 10$     $m < 5$

  • "More than":     $m > 5$

  • "At least":     $x \ge 20$

  • "At most":     $k \le 20$

  • "No more than":     $x \le 8$

  • "No less than":     $x \ge 8$

• Algebraic Rules for Inequalities and Sign Reversal:

  • Multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign:     $-1x \ge 5 \implies x \le -5$     $-5 \ge 2x \implies x \le -\frac{5}{2}$

  • Absolute value inequality restrictions:     $-1|x| > 5 \implies |x| < -5$

  • Because the absolute value of any real number is always non-negative ($|x| \ge 0$), there is no real solution to this inequality.

• Solving Compound Linear Inequalities:

  • Example 1:     $3 \le 2x - 1 \le 9$

  • Add $1$ to all three parts:       $3 + 1 \le 2x \le 9 + 1$       $4 \le 2x \le 10$

  • Divide all three parts by $2$:       $\frac{4}{2} \le x \le \frac{10}{2}$       $2 \le x \le 5$

  • Example 2:     $2 \le 3x - 1 \le 11$

  • Add $1$ to all three parts:       $2 + 1 \le 3x \le 11 + 1$       $3 \le 3x \le 12$

  • Divide all three parts by $3$:       $\frac{3}{3} \le x \le \frac{12}{3}$       $1 \le x \le 4$

Linear Functions, Coordinate Geometry, and Slope Analysis

• Slope-Intercept Form of a Linear Equation:   $y = mx + b$

  • The parameter $m$ represents the slope of the line.

  • The parameter $b$ represents the $y$-intercept.

• Mathematical Interpretations and Formulas for Slope:

  • Slope formula across two coordinate points $(x_1, y_1)$ and $(x_2, y_2)$:     $m = \frac{y_2 - y_1}{x_2 - x_1}$

  • Graphical geometric slope definition:     $m = \frac{\text{rise}}{\text{run}}$

  • Identification from an equation in slope-intercept form:     $m = \text{number beside } x$

  • Real-world contextual meaning:     $\text{Slope} = \text{Average rate} = \text{rate of } y \text{ per } x$

• Classification of Slope Directions:

  • Positive slope ($+ve$): Graph slants upward from left to right ($/$).

  • Negative slope ($-ve$): Graph slants downward from left to right ($\backslash$).

  • Zero slope ($0$): Graph is a perfectly horizontal line ($-$).

  • Undefined slope: Graph is a perfectly vertical line ($|$), where the horizontal displacement is zero ($\text{run} = 0$).

• Intercept Determinations and Practical Meanings:

  • $x$-intercept:

  • Found by setting $y = 0$ and solving for $x$.

  • $y$-intercept:

  • Found by setting $x = 0$ and solving for $y$.

  • Contextually represents the initial value or starting value at $x = 0$.

• Parallel and Perpendicular Line Slope Relationships:

  • Parallel lines ($\parallel$):

  • Slopes are equal:       $m_1 = m_2$

  • Example: If $m_1 = \frac{2}{5}$, then the parallel slope is $m_2 = \frac{2}{5}$.

  • Perpendicular lines ($\perp$):

  • Slopes are opposite reciprocals, having a product of $-1$:       $m_1 \times m_2 = -1$       $m_2 = -\frac{1}{m_1}$

  • Example: If $m_1 = \frac{2}{5}$, then the perpendicular slope is $m_2 = -\frac{5}{2}$.

• Graphical summary of linear slopes and coordinate properties:

Systems of Linear Equations and Solutions Criteria

• Standard Form of a $2 \times 2$ Linear System:   $ax + by = c$   $dx + ey = f$

• Conditions for Number of Solutions:

  • Unique Solution (Exactly $1$ Solution / Intersecting Lines):     $\frac{a}{d} \ne \frac{b}{e}$

  • The lines possess distinct slopes and meet at exactly one coordinate point $(x, y)$.

• No Solution ($0$ Solutions / Parallel Lines / Inconsistent System):   $\frac{a}{d} = \frac{b}{e} \ne \frac{c}{f}$

  • The lines possess identical slopes but different $y$-intercepts, running parallel and never intersecting.

  • Example demonstrating an algebraic contradiction:       $x - 5 = x - 25$       $-5 = -25$

  • Infinitely Many Solutions ($\infty$ Solutions / Coincident Lines / Dependent System):     $\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$

  • The equations represent the same line, intersecting at all points.

• Scientific Calculator Configuration:

  • To solve simultaneous $2 \times 2$ linear equations on a Casio scientific calculator:     $\text{mode } 5, 1$

Quadratic Functions, Discriminant Analysis, and Parabolas

• Standard Form of Quadratic Equations:   $ax^2 + bx + c = 0$

  • To solve quadratic equations on a Casio scientific calculator:     $\text{mode } 5, 3$

• Constructing Equations from Root Parameters:

  • Standard formula based on the sum and product of the roots:     $x^2 - (\text{Sum})x + (\text{Product}) = 0$

  • Example: Given a sum of roots equal to $+5$ and a product of roots equal to $6$:     $x^2 - 5x + 6 = 0$

  • Factoring the quadratic expression confirms the roots:       $(x - 2)(x - 3) = 0 \implies x = 2,\; x = 3$

• Forms of Quadratic Equations and Their Properties:

  • Factored Form:     $y = (x - a)(x - b)$

  • Directly provides the roots, solutions, and $x$-intercepts of the function.

  • Vertex Form:     $y = a(x - h)^2 + k$

  • Directly reveals the vertex coordinates $(h, k)$, the axis of symmetry, and the maximum or minimum value.

• The Discriminant and Nature of Roots:   $\Delta = b^2 - 4ac$

  • If $b^2 - 4ac > 0$ ($+ve$):

  • The equation has $2$ distinct real solutions ($2$ $x$-intercepts).

  • If $b^2 - 4ac = 0$ ($\text{zero}$):

  • The equation has $1$ real solution (a single repeated root where the vertex is on the $x$-axis).

  • If $b^2 - 4ac < 0$ ($-ve$):

  • The equation has no real solutions ($0$ real solutions; the graph does not touch or intersect the $x$-axis).

• Geometric Properties of Parabolas:

  • Vertical orientation:

  • If $a > 0$ ($+ve$): The parabola opens upward ($\cup$), and the vertex is a minimum.

  • If $a < 0$ ($-ve$): The parabola opens downward ($\cap$), and the vertex is a maximum.

  • Vertex Coordinates $(x_v, y_v)$:

  • The $x$-coordinate of the vertex (axis of symmetry or abscissa):       $x_v = -\frac{b}{2a}$

  • The $y$-coordinate of the vertex (maximum/minimum value or ordinate):       $y_v = f(x_v) = f\left(-\frac{b}{2a}\right)$

Function Evaluation, Composition, and Inverses

• Evaluating Functions with Numeric and Algebraic Inputs:

  • Given the linear function:     $f(x) = 2x + 5$

  • Evaluating for positive integer $x = 4$:     $f(4) = 2(4) + 5 = 8 + 5 = 13$

  • Evaluating for negative integer $x = -5$:     $f(-5) = 2(-5) + 5 = -10 + 5 = -5$

  • Evaluating for variable input $m$:     $f(m) = 2m + 5$

  • Evaluating for variable input $k$:     $f(k) = 2k + 5$

  • Evaluating for scaled algebraic input $2a$:     $f(2a) = 2(2a) + 5 = 4a + 5$

• Composite Functions:

  • Given the definitions:     $f(x) = 2x + 1$     $g(x) = x^2$

  • Evaluating $f(g(2))$:

  • Evaluate inner function $g(2)$ first:       $g(2) = (2)^2 = 4$

  • Substitute this value into $f(x)$:       $f(4) = 2(4) + 1 = 8 + 1 = 9$

  • Evaluating $g(f(2))$:

  • Evaluate inner function $f(2)$ first:       $f(2) = 2(2) + 1 = 4 + 1 = 5$

  • Substitute this value into $g(x)$:       $g(5) = (5)^2 = 25$

• Determining the Inverse Function ($f^{-1}(x)$):

  • Algorithmic steps:

  1. Switch the positions of variables $x$ and $y$.

  2. Solve the newly formed equation for $y$.

  • Worked application for $y = 2x + 5$:

  • Step 1 (Switch $x$ and $y$):       $x = 2y + 5$

  • Step 2 (Solve for $y$):       $x - 5 = 2y$       $y = \frac{x - 5}{2}$

  • Express in inverse function notation:       $f^{-1}(x) = \frac{x - 5}{2}$

Polynomial Degrees, Shapes, and End Behavior

• Correlation of Polynomial Degree and Leading Coefficient to Curve Shape:

  • Degree 1 (Linear Polynomials):

  • $+x$: Linear graph sloping upward to the right ($/$).

  • $-x$: Linear graph sloping downward to the right ($\backslash$).

  • Degree 2 (Quadratic Polynomials):

  • $+x^2$: Parabola opening upward ($\cup$); both ends point up ("end up").

  • $-x^2$: Parabola opening downward ($\cap$); both ends point down ("end down").

  • Degree 3 (Cubic Polynomials):

  • $+x^3$: Shape resembling an "$N$"; starts downward on the left and ends upward on the right ("end up").

  • $-x^3$: Shape resembling an inverted "$N$"; starts upward on the left and ends downward on the right ("end down").

  • Degree 4 (Quartic Polynomials):

  • $+x^4$: Shape resembling a "$W$"; both ends point upward ("end up").

  • $-x^4$: Shape resembling an "$M$"; both ends point downward ("end down").

• Reference diagram showing polynomial end behaviors and the inverse derivation process:

Rational Expressions and the Remainder Theorem

• Domain Restrictions for Rational Functions:

  • A rational algebraic fraction is mathematically undefined when its denominator equals zero:     $\text{Denominator} = 0$

  • Application to rational expression:     $\frac{x + 2}{x - 5}$

  • Set the denominator equal to zero:       $x - 5 = 0 \implies x = 5$

  • The expression is undefined at $x = 5$.

• Polynomial Division and the Remainder Theorem:

  • The Remainder Theorem states that when a polynomial $P(x)$ is divided by a linear factor of the form $x - c$, the remainder $R$ equals the polynomial evaluated at $c$:     $x - c = 0 \implies x = c$     $R = P(c)$

  • Application to polynomial $P(x) = x^2 - 5x + 7$ when divided by $x - c$:     $R = c^2 - 5c + 7$

Coordinate Circle Equations and Geometric Angle Theorems

• Equations of Circles in the Cartesian Plane:

  • Standard Form:     $(x - h)^2 + (y - k)^2 = r^2$

  • Coordinates of the center:       $\text{Center} = (h, k)$

  • Radius of the circle:       $\text{Radius} = r = \sqrt{r^2}$

  • General Expanded Form:     $x^2 + ax + y^2 + by = c$

  • Coordinates of the center obtained by completing the square:       $\text{Center} = \left(-\frac{a}{2}, -\frac{b}{2}\right)$

  • Formula for the radius:       $\text{Radius} = \sqrt{\left(-\frac{a}{2}\right)^2 + \left(-\frac{b}{2}\right)^2 + c}$

• Angle Relationships in Geometry:

  • Sum of interior angles in any triangle:     $\text{Sum} = 180^\circ$

  • Equilateral triangle properties:

  • All three interior angles are identical:       $60^\circ + 60^\circ + 60^\circ = 180^\circ$

  • Supplementary angles on a straight line:

  • Angles forming a linear pair sum to $180^\circ$.

  • For an angle measuring $80^\circ$:       $180^\circ - 80^\circ = 100^\circ$

  • For an angle measuring $120^\circ$:       $180^\circ - 120^\circ = 60^\circ$

Trigonometric Waves, Right Triangles, and Identities

• Trigonometric Functions and Periodic Waveforms:

  • Sine function waveform:     $y = a\sin(bx)$

  • Cosine function waveform:     $y = a\cos(bx)$

  • Amplitude formula:     $\text{Amplitude} = |a|$

  • Period formula:     $\text{Period} = \frac{2\pi}{b}$

• Right-Triangle Trigonometry (SOH CAH TOA):

  • Pythagorean Theorem connecting hypotenuse $c$ with legs $a$ and $b$:     $c = \sqrt{a^2 + b^2}$     $a = \sqrt{c^2 - b^2}$     $b = \sqrt{c^2 - a^2}$

  • Primary trigonometric ratios:

  • Sine (Opposite over Hypotenuse):       $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

  • Cosine (Adjacent over Hypotenuse):       $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • Tangent (Opposite over Adjacent):       $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Cofunction Identity for Complementary Angles:

  • The sine of an angle is equivalent to the cosine of its complementary angle:     $\sin(x) = \cos(y) \iff x + y = 90^\circ$     $\sin(x) = \cos(90^\circ - x)$

Exponential Growth, Logarithms, and Exponent Rules

• Exponential Growth and Decay Model:   $A = P(1 \pm r)^t$

  • $A$ represents the final accrued amount.

  • $P$ represents the initial baseline principal amount.

  • $r$ represents the fractional rate of growth ($+$, increase) or decay ($–$, decrease).

  • $t$ represents the number of compounding time intervals.

• Foundations and Conversions of Logarithms:

  • Equivalence between exponential and logarithmic forms:     $b^y = x \iff \log_b(x) = y$

  • Evaluated examples:     $2^3 = 8 \iff \log_2(8) = 3$     $3^2 = 9 \iff \log_3(9) = 2$

• Algebraic Properties of Logarithms:

  • Product rule of logarithms:     $\log(x) + \log(y) = \log(xy)$

  • Quotient rule of logarithms:     $\log(x) - \log(y) = \log\left(\frac{x}{y}\right)$

  • Combined condensation of sums and differences:     $\log(x) + \log(y) - \log(z) = \log\left(\frac{xy}{z}\right)$

• Exponent and Radical Operational Rules:

  • Converting fractional exponents to radical notation:     $x^{\frac{a}{b}} = \sqrt[b]{x^a}$

  • Application:       $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$

  • Product rule for radicals:     $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

  • Quotient rule for radicals:     $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Financial Percentages, Statistics, and Probability

• Discount Calculations and Percentage Change:

  • Discounted Price Evaluation:

  • Context: An article of clothing with an original price of \\$150 is discounted by $20\%$.

  • The remaining percentage paid by the purchaser is:       $100\% - 20\% = 80\%$

  • Calculating the final cost:       $80\% \times 150 = \frac{80}{100} \times 150 = 0.80 \times 150 = 120$

  • The final price after discount is \\$120.

  • Percentage Change Formula:     $\%\,\text{Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100$

• Measures of Central Tendency and Data Spread:

  • Mean (Arithmetic Average):     $\text{Mean} = \frac{\text{Sum}}{\text{Number}}$

  • Mode:     $\text{Mode} = \text{The most repeated value in a dataset}$

  • Median:     $\text{Median} = \text{The middle value after arranging the data in ascending numerical order}$

  • Range:     $\text{Range} = \text{Greatest} - \text{Smallest} = G - S$

• Theoretical Probability Ratio:

  • Definition:     $\text{Probability} = \frac{\text{Part}}{\text{Total}}$