Precalculus Foundations: Algebra, Functions, and Trigonometry Study Notes

Unit 1: Foundations for Precalculus

This unit reviews and strengthens essential precalculus skills, focusing on algebra, functions, and trigonometry. The following sections provide structured explanations, definitions, examples, and key equations for each topic.

Section 1.1: Fraction Arithmetic

• Order of Operations: Use the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) to simplify numerical expressions involving parentheses, exponents, and fractions. Example: Simplify $2 + 3 \times (4^2 - 6) \div 2$ Solution: $2 + 3 \times (16 - 6) \div 2 = 2 + 3 \times 10 \div 2 = 2 + 30 \div 2 = 2 + 15 = 17$

• Multiplying and Dividing Fractions: Multiply numerators and denominators directly; to divide, multiply by the reciprocal. Example: $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$; $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

• Adding and Subtracting Fractions: Use a common denominator, then add or subtract numerators. Example: $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$

Section 1.2: Exponents, Roots, & Intervals

• Laws of Exponents (Integer and Fractional): Apply rules such as $a^m \times a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$, $a^0 = 1$, $a^{-n} = \frac{1}{a^n}$, $a^{1/n} = \sqrt[n]{a}$. Example: $2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$ $16^{1/2} = \sqrt{16} = 4$

• Interval Notation: Use interval notation to represent sets of numbers. Example: $[2, 5)$ represents all real numbers $x$ such that $2 \leq x < 5$.

Section 1.3: Functions and Lines

• Function Notation: $f(x)$ denotes the value of function $f$ at $x$. Example: If $f(x) = 2x + 3$, then $f(4) = 2 \times 4 + 3 = 11$.

• Domain and Range: The domain is the set of all possible input values; the range is the set of all possible output values. Example: For $f(x) = \sqrt{x}$, domain: $[0, \infty)$, range: $[0, \infty)$.

• Slope and Linear Equations: The slope $m$ of a line through $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$. The equation of a line: $y = mx + b$ or point-slope form $y - y_1 = m(x - x_1)$. Example: Find the equation of a line through $(1,2)$ with slope $3$:$y - 2 = 3(x - 1) \implies y = 3x - 1$.

Section 1.4: Function Symmetry and Translation

• Even and Odd Functions: A function is even if $f(-x) = f(x)$ (symmetric about the y-axis), odd if $f(-x) = -f(x)$ (symmetric about the origin). Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

• Basic Functions: $y = x^2$, $y = x^3$, $y = \frac{1}{x}$, $y = \frac{1}{x^2}$, $y = \sqrt{x}$, $y = \sqrt[3]{x}$, $y = |x|$ are fundamental parent functions. Example: The graph of $y = |x|$ is a "V" shape opening upwards.

• Graphing Functions: Identify and state the domain and range of parent functions; graph vertical and horizontal shifts. Example: $y = (x - 2)^2 + 3$ is $y = x^2$ shifted right by 2 and up by 3.

Section 1.5: The Unit Circle and Trigonometric Functions

• Degrees and Radians: $180^\circ = \pi$ radians. To convert: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$. Example: $90^\circ = 90 \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.

• Sine and Cosine: For angle $\theta$ on the unit circle, $\sin \theta$ is the y-coordinate, $\cos \theta$ is the x-coordinate. Example: $\sin 0 = 0$, $\cos 0 = 1$.

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$.

• Six Trigonometric Functions: Sine, cosine, tangent, cotangent, secant, cosecant. $\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\csc \theta = \frac{1}{\sin \theta}$.

• Special Angles: Know values for $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$ in all quadrants.

Section 1.6: Translating and Transforming Trigonometric Functions; Solving Trig Equations

• Graphing Trigonometric Functions: Graph sine, cosine, tangent, cotangent, secant, cosecant, noting amplitude, period, phase shift, and vertical shift. Example: $y = 2\sin(x - \frac{\pi}{4}) + 1$ has amplitude 2, shifted right by $\frac{\pi}{4}$, up by 1.

• Solving Trigonometric Equations: Find exact or approximate solutions for equations like $\sin x = \frac{1}{2}$ within a given interval. Example: $\sin x = \frac{1}{2}$, $x = \frac{\pi}{6}, \frac{5\pi}{6}$ (in $[0, \pi]$).

Section 1.7: Changing the Form of Algebraic Expressions (Factoring)

• Factoring: Identify and extract the greatest common factor (GCF) from algebraic expressions. Example: $6x^2 + 9x = 3x(2x + 3)$

• Special Forms: Factor difference of squares: $a^2 - b^2 = (a - b)(a + b)$; sum/difference of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

• Quadratic Expressions: Factor $ax^2 + bx + c$ using grouping or the quadratic formula. Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$

• Simplifying Rational Expressions: Factor numerators and denominators, reduce common factors. Example: $\frac{x^2 - 9}{x^2 - 3x} = \frac{(x + 3)(x - 3)}{x(x - 3)} = \frac{x + 3}{x}$ for $x \neq 0, 3$