Precalculus Unit 1: Foundations, Functions, and Trigonometry Study Notes

Unit 1: Foundations of Precalculus

This unit introduces essential precalculus skills, focusing on algebraic operations, functions, exponents, trigonometry, and problem-solving techniques. Mastery of these topics is crucial for success in calculus and higher mathematics.

Section 1.1: Fractions & Parentheses

Understanding how to manipulate expressions with fractions and parentheses is fundamental in algebra.

• Order of Operations: Use the correct sequence (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) to simplify expressions.

• Parentheses, Exponents, and Fractions: Simplify expressions by resolving parentheses first, then exponents, followed by operations with fractions.

• Multiplying and Dividing Fractions: Multiply numerators and denominators directly; for division, multiply by the reciprocal.

• Adding/Subtracting Fractions: Use a common denominator, combine numerators, and simplify.

Example: Simplify $\frac{2}{3} + \frac{4}{9}$.

Find a common denominator (9): $\frac{2}{3} = \frac{6}{9}$, so $\frac{6}{9} + \frac{4}{9} = \frac{10}{9}$.

Section 1.2: Exponents, Roots, & Intervals

Exponents and roots are used to represent repeated multiplication and extraction of roots, respectively. Intervals describe sets of numbers on the real number line.

• Exponent Laws: Apply rules such as $a^m \cdot a^n = a^{m+n}$, $(a^m)^n = a^{mn}$, and $a^0 = 1$.

• Fractional Exponents: $a^{1/n} = \sqrt[n]{a}$, $a^{m/n} = (\sqrt[n]{a})^m$.

• Interval Notation: Use brackets and parentheses to describe intervals, e.g., $[a, b]$ for closed, $(a, b)$ for open intervals.

Example: Simplify $16^{3/4}$.

$16^{3/4} = (16^{1/4})^3 = (2)^3 = 8$.

Section 1.3: Functions and Lines

Functions describe relationships between variables. Linear functions are represented by straight lines.

• Function Notation: $f(x)$ represents the output when $x$ is the input.

• Domain and Range: The domain is the set of possible inputs; the range is the set of possible outputs.

• Slope and Graphing: The slope $m$ of a line measures its steepness; the equation $y = mx + b$ describes a line.

• Equation from Point and Slope: $y - y_1 = m(x - x_1)$.

• Parallel and Perpendicular Lines: Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Piecewise Functions: Functions defined by different expressions over different intervals.

Example: Find the equation of a line passing through $(2, 3)$ with slope $4$.

$y - 3 = 4(x - 2)$

Section 1.4: Function Symmetry and Translation

Functions can exhibit symmetry and can be transformed by shifting, stretching, or reflecting their graphs.

• Even and Odd Functions: Even: $f(-x) = f(x)$; Odd: $f(-x) = -f(x)$.

• Basic Parent Functions: Examples include $y = x$, $y = x^2$, $y = \sqrt{x}$, $y = \frac{1}{x}$.

• Domain and Range: Identify for each parent function.

• Vertical and Horizontal Shifts: $f(x) + c$ shifts up, $f(x - h)$ shifts right.

• Reflections: $-f(x)$ reflects over the $x$-axis; $f(-x)$ reflects over the $y$-axis.

Example: $f(x) = (x - 2)^2 + 3$ is a parabola shifted right by 2 and up by 3.

Section 1.5: The Unit Circle and Trig Functions

The unit circle is a fundamental tool for understanding trigonometric functions and their values.

• Degrees and Radians: $360^\circ = 2\pi$ radians.

• Sine and Cosine: For angle $\theta$, $\sin \theta$ and $\cos \theta$ are the $y$ and $x$ coordinates on the unit circle.

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

• Special Angles: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$.

• Reference Angles: Used to find trig values for any angle.

Example: $\sin 30^\circ = \frac{1}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$.

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

Trigonometric functions can be shifted and stretched, and their equations solved for specific values.

• Basic Trig Functions: Sine, cosine, tangent, cotangent, secant, cosecant.

• Transformations: Amplitude, period, phase shift, vertical shift.

• Solving Trig Equations: Find exact or approximate solutions within a given interval.

Example: Solve $\sin x = \frac{1}{2}$ for $x$ in $[0, 2\pi]$.

Solutions: $x = \frac{\pi}{6}, \frac{5\pi}{6}$.

Section 1.7: Changing the Form of a Function (Factoring)

Factoring is the process of rewriting expressions as products of simpler expressions.

• Greatest Common Factor (GCF): Factor out the largest common factor.

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

• Quadratic Expressions: Factor into $(x - r_1)(x - r_2)$ where $r_1, r_2$ are roots.

• Grouping Method: Group terms to factor polynomials with four or more terms.

• Polynomial Long Division: Divide polynomials to find factors.

Example: Factor $x^2 - 9 = (x - 3)(x + 3)$.

Section 1.8: Solving Equations (Linear, Quadratic, Other Types)

Solving equations involves finding values of variables that satisfy given algebraic statements.

• Linear Equations: $ax + b = 0$; solve for $x$.

• Quadratic Equations: $ax^2 + bx + c = 0$; use factoring, completing the square, or quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• Systems of Equations: Solve two equations simultaneously using substitution or graphing.

• Extraneous Solutions: Check for solutions that do not satisfy the original equation.

• Real-World Problems: Apply equations to model and solve practical scenarios.

Example: Solve $2x + 3 = 7$.

$2x = 4 \Rightarrow x = 2$.

Section 1.9: Solving a System of Equations

Systems of equations involve finding values that satisfy multiple equations simultaneously.

• Substitution Method: Solve one equation for a variable and substitute into the other.

• Graphing Method: Plot both equations and find the intersection point.

Example: Solve $x + y = 5$ and $x - y = 1$.

Add: $2x = 6 \Rightarrow x = 3$, then $y = 2$.

Table: Key Concepts and Methods in Precalculus Unit 1

Topic	Key Concepts	Example
Fractions & Parentheses	Order of operations, simplifying fractions	$\frac{2}{3} + \frac{4}{9} = \frac{10}{9}$
Exponents & Roots	Exponent laws, fractional exponents	$16^{3/4} = 8$
Functions & Lines	Function notation, slope, graphing	$y - 3 = 4(x - 2)$
Trig Functions & Unit Circle	Degrees/radians, sine/cosine, special angles	$\sin 30^\circ = \frac{1}{2}$
Factoring	GCF, difference of squares, grouping	$x^2 - 9 = (x - 3)(x + 3)$
Solving Equations	Linear, quadratic, systems	$x = 2$ for $2x + 3 = 7$