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 .

Find a common denominator (9): , so .

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 , , and .

• Fractional Exponents: , .

• Interval Notation: Use brackets and parentheses to describe intervals, e.g., for closed, for open intervals.

Example: Simplify .

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Section 1.3: Functions and Lines

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

• Function Notation: represents the output when 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 of a line measures its steepness; the equation describes a line.

• Equation from Point and Slope: .

• 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 with slope .

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: ; Odd: .

• Basic Parent Functions: Examples include , , , .

• Domain and Range: Identify for each parent function.

• Vertical and Horizontal Shifts: shifts up, shifts right.

• Reflections: reflects over the -axis; reflects over the -axis.

Example: 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: radians.

• Sine and Cosine: For angle , and are the and coordinates on the unit circle.

• Pythagorean Identity: .

• Special Angles: .

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

Example: , .

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 for in .

Solutions: .

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: ; Sum/difference of cubes: .

• Quadratic Expressions: Factor into where are roots.

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

• Polynomial Long Division: Divide polynomials to find factors.

Example: Factor .

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

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

• Linear Equations: ; solve for .

• Quadratic Equations: ; use factoring, completing the square, or quadratic formula: .

• 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 .

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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 and .

Add: , then .

Table: Key Concepts and Methods in Precalculus Unit 1