Math 1030 Pre-Calculus Final Review Vocabulary

General Examination Guidelines and Requirements

• Required Materials: Students are permitted to use only a $TI-30XIIS$ calculator. Other types of calculators (including graphing calculators), cell phones, and smart watches are strictly prohibited.

• Documentation: All work must be shown clearly to receive full credit.

• Answer Formatting: Instructions specify that all answers must be simplified. Numerical answers should remain in exact form (fractions or square roots) and should not be converted to decimals unless specifically requested.

Foundations of Relations and Functions

• Relation Definition: A relation is examined through the set of ordered pairs: ${(-1, 4), (0, 1), (1, 4), (0, 1), (2, 4)}$.     * Determining if a relation is a function requires verifying if every input has exactly one output.     * Domain: The set of all possible input values (x-coordinates).     * Range: The set of all possible output values (y-coordinates).

• Piecewise Functions: Evaluation of functions defined by different formulas over specific intervals.     * Example: $g(x) = \begin{cases} \sqrt{x^2 + 4} & \text{if } x \leq 3 \ 1 - x & \text{if } x > 3 \end{cases}$.     * Evaluation points include $g(-3)$, $g(4)$, and boundary point $g(3)$.

• Domain Determination: Analysis of function domains, expressed in interval notation:     * Rational functions such as $f(x) = \frac{1 - x}{8 - 4x}$ (exclude values where denominator is zero).     * Radical functions such as $f(x) = \sqrt{x - 4}$ (radicand must be non-negative).     * Polynomial functions such as $f(x) = 7x^4 - 2x^3 + \sqrt{2}$ (domain is all real numbers).

Algebraic Operations and Function Composition

• Function Combinations: Given $f(x) = 3x^2 - 1$ and $g(x) = 1 - 2x$, find formulas for:     * Sum: $f(x) + g(x)$     * Difference: $f(x) - g(x)$     * Scalar Multiples and Sums: $3f(x) + 2g(x)$     * Product: $f(x)g(x)$

• Function Composition: Given $f(x) = 1 - x^2$ and $g(x) = 3x + 2$, determine:     * $(f \circ g)(x) = f(g(x))$     * $(g \circ f)(x) = g(f(x))$     * $(g \circ g)(x) = g(g(x))$

• Solving Function Equations:     * Finding $x$ such that $f(x) = 7$ for $f(x) = 2x - 9$.     * Finding $t$ such that $g(t) = 0$ for $g(t) = t^2 - t - 20$.

• Difference Quotient: Calculation and simplification of $\frac{f(x+h) - f(x)}{h}$ for functions like $f(x) = 2x^2 + 1$ and $f(x) = 4x - 1$.

Graphical Analysis and Inverse Functions

• Feature Extraction from Graphs:     * Identification of Domain and Range.     * Intervals of positivity (f(x) > 0) and negativity (f(x) < 0).     * Intercepts ($x$-intercepts and $y$-intercepts).     * Intervals of behavior: Increasing, decreasing, or constant.     * Zeros: Points where $f(x) = 0$.     * One-to-one property testing (Horizontal Line Test).

• Inverse Functions ($f^{-1}$):     * Existence: $f^{-1}$ exists if the function is one-to-one.     * Solving for the inverse analytically for linear functions $f(x) = 2x + 7$ and rational functions $f(x) = \frac{x + 1}{3 - x}$.     * Root-based inverses like $f(x) = \sqrt{2x + 9}$ and $f(x) = \sqrt[3]{1 - x}$.     * Graphical relationships: $f^{-1}(3)$ corresponds to finding $x$ when $f(x) = 3$. Solving $f^{-1}(y) = 2$ corresponds to finding $y = f(2)$.

Parent Functions and Graph Transformations

• Identification and Transformation Steps: Starting from a parent function $f(x)$, describe reflections, shifts, and stretches to obtain $g(x)$.     * Quadratic: $g(x) = -x^2 + 4$ (Parent $f(x) = x^2$; reflection over x-axis, vertical shift up $4$).     * Square Root: $g(x) = -\sqrt{x} - 3$ (Parent $f(x) = \sqrt{x}$; reflection over x-axis, vertical shift down $3$).     * Absolute Value: $g(x) = -|x + 3| - 2$ (Parent $f(x) = |x|$; reflection over x-axis, horizontal shift left $3$, vertical shift down $2$).

Quadratic Functions and Polynomials

• Quadratic Analysis: For $f(x) = x^2 - 6x - 7$:     * Vertex calculation $(h, k)$.     * Conversion to Standard (Vertex) Form.     * Identification of intercepts and sketching.

• Parabola Construction: Finding standard form using a vertex and a specific point. Trials include:     * Vertex: $(1, -3)$, Point: $(2, -1)$.     * Vertex: $(2, -3)$, Point: $(3, 0)$.     * Vertex: $(5, 12)$, Point: $(7, 15)$.

• Polynomial Characteristics:     * Rational Zeros Theorem: Listing all possible rational zeros for $f(x) = 3x^3 + 4x^2 - 7x + 6$ using $\pm \frac{p}{q}$.     * Degree and Multiplicity: For $f(x) = (x + 2)^3 x(x - 1)$, identifying degree and behavior at roots.     * Factor Theorem: Testing if $(x + 1)$ is a factor of $f(x) = 2x^3 + x^2 - 2x - 1$ by evaluating $f(-1)$.

• Modeling and Optimization: Application of quadratic vertex to real-world costs.     * Cost function: $C(x) = 120000 - 0.2x + 0.001x^2$.     * Minimum cost occurs at the vertex $x = \frac{-b}{2a}$.

Rational Functions

• Domain Constraints: Determining where the denominator is non-zero, e.g., $f(x) = \frac{2x+1}{2x-7}$.

• Advanced Domain: Solving inequalities for radical-rational hybrids like $f(x) = \sqrt{x^2 - 6x + 5}$.

• Comprehensive Graphing: For $f(x) = \frac{x + 1}{x - 2}$ and $f(x) = \frac{4x}{x^2 - 4}$:     * Vertical Asymptotes (zeros of denominator).     * Horizontal Asymptotes (comparison of numerator/denominator degrees).     * Intercepts, Domain, and Range determination.

Exponential and Logarithmic Functions

• Calculator Evaluations:     * $\ln(3.43 + 50.27)$     * $e^2 + 3e$     * $\ln(\frac{\pi}{4})$     * $\frac{10 \ln(5)}{1 + 3 \ln(4)}$

• Exact Value Evaluations (No Calculator):     * $\ln(e^{2+7x}) = 2 + 7x$     * $\log_{32}(4)$     * $\log_{4}(64)$

• Converting Forms:     * Logarithmic to Exponential: $\log_5(25) = 2 \implies 5^2 = 25$.     * Exponential to Logarithmic: $3^4 = 81 \implies \log_3(81) = 4$.

• Logarithmic Properties:     * Condensing: Writing expressions as a single logarithm using product, quotient, and power rules.     * Expanding: Breaking down expressions like $\log_{10}\left(\frac{x^2 \sqrt{y}}{100z^7}\right)$ into sums and differences.

• Equations Solving (Exact Answers):     * Exponential: $3(10^{2x-3}) = 5$, $7^{5x} = 3^{x-2}$, $7^{2x-1} = 6$.     * Logarithmic: $\ln^3(x - 2) = 1$, $\log_4(x) + \log_4(x - 3) = 1$, $\log_2(x) - \log_2(x - 1) = 2$.

• Growth Applications: Bacteria population model $P(t) = 200e^{0.0293t}$.     * Initial population ($t=0$).     * Population after specific hours ($t=3$, $t=5$).     * Doubling time: solve $400 = 200e^{0.0293t}$ for $t$.

• Function Characterization: Given $g(x) = e^{x-1} + 2$ or $g(x) = \ln(x + 2) - 2$:     * Determine parent function, transformations, intercepts, domain, range, and asymptotic behavior.

Trigonometric Fundamentals

• Angle Conversion:     * Radians to Degrees: Multiply by $\frac{180}{\pi}$. Example: $\theta = \frac{\pi}{12}$, $\theta = \frac{9\pi}{5}$, $\theta = \frac{5}{12}$.     * Degrees to Radians: Multiply by $\frac{\pi}{180}$.

• Trigonometric Values via Calculator:     * Calculating $\sin(65^\circ)$, $\cos(\frac{5\pi}{13})$, $\sec(200^\circ)$, $\cot(22)$.

• Unit Circle and Standard Position:     * Sketching angles like $210^\circ$ or $-\frac{7\pi}{12}$.     * Determining quadrants and finding positive/negative coterminal angles.

• Arc Length: Use formula $s = r\theta$ (where $\theta$ is in radians).     * $r = 10.267\,\text{in}$, $50^\circ$     * $r = 5\,\text{ft}$, $\theta = 3$     * $r = 20\,\text{m}$, $240^\circ$

• Six Trigonometric Functions on the Plane:     * Given a point $(x, y)$ on the terminal side, $\sin \theta = \frac{y}{r}$, $\cos \theta = \frac{x}{r}$, $\tan \theta = \frac{y}{x}$, where $r = \sqrt{x^2 + y^2}$.     * Example points: $(-3, 2)$, $(5, -13)$, $(-5, 8)$, $(-7, -7)$.

Advanced Trigonometry and Identities

• Constrained Trig Values: Finding exact values given specific conditions:     * $\tan(\theta) = -\frac{12}{5}$ and \cos(\theta) < 0.     * $\cos(\theta) = -\frac{5}{13}$ and \tan(\theta) < 0.     * $\tan(\theta) = \frac{15}{8}$ and \csc(\theta) < 0.

• Applications: Finding pole heights using right triangle trigonometry ($21\,\text{ft}$ base, $64^\circ$ angle).

• Reference Angles ((\theta')): Determining $\theta'$ for angles like $135^\circ$, $-\frac{\pi}{6}$, $330^\circ$.

• Inverse Trigonometric Composition: Finding the exact value of $\cos(\tan^{-1}(-\frac{2}{5}))$.

• Graphing Sine, Cosine, and Tangent:     * Function forms: $f(x) = A \sin(Bx - C) + D$ or $f(x) = A \cos(Bx - C) + D$.     * Amplitude ($|A|$), Period ($\frac{2\pi}{B}$), and Phase Shifts.     * Example: $f(x) = 2 \sin(\pi x) + 1$, $f(x) = 2 \tan(4x) + 2$.

• Identity Simplification:     * Utilizing Pythagorean identities: $\sin^2 t + \cos^2 t = 1$.     * Simplifying $\sin \theta(\csc \theta - \sin \theta)$ or $\sin \beta \tan \beta + \cos \beta$.

• Trigonometric Substitution: Rewriting algebraic radicals like $\sqrt{9 - x^2}$ using $x = 3 \sin \theta$.

• Double Angle and Sum/Difference Formulas:     * Double Angle: $\sin(2\theta) = 2 \sin \theta \cos \theta$     * Sum/Difference: Calculating exact values for $\sin(75^\circ)$ or $\cos(285^\circ)$ using $(45^\circ + 30^\circ)$ or $(45^\circ + 60^\circ)$.

Trigonometric Equations and Triangle Solving

• Interval Solving ($0 \leq t \leq 2\pi$):     * $1 - 3 \tan^2 t = 0$     * $4 \cos^2 t - 1 = 0$

• General Solutions: Finding all solutions for $2 \sin(4\theta) + 1 = 0$ in radians.

• Law of Sines and Law of Cosines: Solving non-right triangles $ABC$.     * Case 1: $A = 48^\circ$, $B = 99^\circ$, $b = 13$ (Law of Sines).     * Case 2: $c = 12$, $b = 9$, $A = 25^\circ$ (Law of Cosines).