Algebra Essentials

Polynomials, Rational Expressions, and Equations

This section covers foundational algebraic skills necessary for success in precalculus, including operations with polynomials, rational expressions, and solving equations.

• Polynomials: Expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

• Synthetic Division: A shortcut method for dividing a polynomial by a linear binomial of the form (x - c).

• Rational Expressions: Fractions where the numerator and/or denominator are polynomials.

• Solving Equations: Techniques for finding the values of variables that satisfy algebraic equations.

• Interval Notation: A way to describe sets of numbers, often used to express the solution sets of inequalities.

• Solving Inequalities: Finding all values of a variable that make an inequality true.

Example: Solve the inequality .

• Factor:

• Solution:

Analytic Geometry

Distance, Midpoint, Lines, and Circles

Analytic geometry connects algebra and geometry, allowing geometric problems to be solved using algebraic equations.

• Distance Formula: Calculates the distance between two points and :

• Midpoint Formula: Finds the midpoint between two points:

• Lines: The equation of a line in slope-intercept form is .

• Circles: The standard equation is , where is the center and is the radius.

Functions and Their Graphs

Definition, Properties, and Graphing

Functions are fundamental objects in mathematics, describing relationships between variables.

• Function: A relation where each input has exactly one output.

• Graph of a Function: The set of all points in the plane.

• Properties: Domain, range, intercepts, symmetry, increasing/decreasing behavior.

• Library of Functions: Common basic functions such as linear, quadratic, cubic, absolute value, square root, and reciprocal functions.

• Graphing Techniques: Shifting, reflecting, stretching, and compressing graphs.

Example: The graph of is a 'V' shape with vertex at the origin.

Polynomial and Rational Functions

Graphing, Properties, and Inequalities

Polynomial and rational functions are key classes of functions with distinct behaviors and graphing techniques.

• Polynomial Functions: Functions of the form .

• Graphing Polynomials: Analyze end behavior, intercepts, and turning points.

• Rational Functions: Ratios of two polynomials, .

• Properties: Vertical/horizontal asymptotes, holes, intercepts.

• Polynomial & Rational Inequalities: Solve by finding zeros and testing intervals.

Example: Find the vertical asymptote of : .

Exponential and Logarithmic Functions

Definitions, Properties, and Equations

Exponential and logarithmic functions model growth, decay, and many real-world phenomena.

• Exponential Function: , where , .

• Logarithmic Function: , the inverse of the exponential function.

• Properties: Laws of exponents and logarithms, such as .

• Equations: Techniques for solving exponential and logarithmic equations.

Example: Solve . Solution: .

Trigonometric Functions

Angles, Unit Circle, and Graphs

Trigonometry studies the relationships between angles and sides of triangles, and extends to periodic functions.

• Angles: Measured in degrees or radians.

• Unit Circle: A circle of radius 1 centered at the origin, used to define sine and cosine for all real numbers.

• Trigonometric Functions: Sine, cosine, tangent, and their reciprocals.

• Graphs: Sine and cosine functions are periodic with amplitude and period.

• Other Trig Functions: Tangent, cotangent, secant, cosecant.

Example: The graph of has period and amplitude 1.

Analytic Trigonometry

Inverse Functions and Trigonometric Equations

This section focuses on solving trigonometric equations and understanding inverse trigonometric functions.

• Inverse Sine, Cosine, Tangent: Functions that reverse the effect of the original trigonometric functions.

• Trigonometric Equations: Equations involving trigonometric functions, solved using identities and algebraic techniques.

Example: Solve for in : .

Course Structure and Assessment

Review, Tests, and Final Exam

The course includes regular reviews, five major tests, and a comprehensive final exam. Labs are scheduled weekly to reinforce concepts through practice and application.

• Review Sessions: Held before each test and the final exam to consolidate learning.

• Tests: Assess understanding of recent topics.

• Final Exam: Cumulative assessment covering all course material.

Summary Table: Major Precalculus Topics