Algebra 2 Fundamentals equips students with a robust understanding of quadratic and rational functions, emphasizing practical application through graphical analysis and algebraic manipulation. Students will master techniques for identifying key features of functions and solving real-world problems, culminating in the ability to analyze and graph complex rational expressions.

Quadratic Functions: Foundations

• Identify the axis of symmetry—the line that divides the parabola into two symmetrical halves
• Determine the vertex—the turning point of the parabola, crucial for understanding function behavior
• Find x-intercepts—the points where the graph crosses the x-axis, representing real roots
• Locate the y-intercept—the point where the graph intersects the y-axis, providing a constant term

Analyzing Quadratic Graphs

• Determine increasing and decreasing intervals—regions where the function's value rises or falls
• Define domain and range—the set of all possible input (x) and output (y) values, respectively
• Understand the relationship between a quadratic equation and its corresponding parabolic graph

Rational Functions: Introduction

• Define rational functions as polynomial over polynomial expressions
• Recognize that rational functions are defined by asymptotes rather than axes of symmetry

Simplifying Rational Expressions

• Factor polynomials in the numerator and denominator to identify common factors
• Cancel common factors to reduce the expression to its simplest form—essential for further analysis

Operations with Rational Functions

• Multiply rational expressions by factoring, canceling common factors, and multiplying straight across
• Divide rational expressions by inverting the second fraction and multiplying

Asymptotes and Graphing

• Identify vertical asymptotes by finding values that make the denominator zero
• Determine horizontal asymptotes by examining the degrees of the numerator and denominator
• Locate holes in the graph by identifying canceled factors—points where the function is undefined but does not have an asymptote
• Use asymptotes and intercepts to sketch the graph of a rational function

Advanced Graphing Techniques

• Analyze the behavior of rational functions near asymptotes—how the graph approaches these lines
• Identify key features such as intercepts, asymptotes, and holes to accurately graph rational functions