Chapter 3 Part 6 | Rational Functions & Their Graphs | Asymptotes, Domain & Intercepts

📘 Ethiopian University Freshman Mathematics – Chapter 3 Part 6: Rational Functions and Their Graphs. In this lesson, we study rational functions and develop a systematic method for understanding, analyzing, and sketching their graphs. You will learn the definition of a rational function in the form �, where � and � are polynomial functions and �, and understand why the denominator determines restrictions on the domain. We explain how to find the domain of a rational function, identify excluded values, determine real zeros, find x-intercepts and y-intercepts, and analyze the behavior of a function near important values. The lesson also introduces the meaning of �, �, �, and �, together with the corresponding behavior of �. These ideas are essential for understanding limits informally and for accurately sketching rational-function graphs. A major focus of this chapter section is asymptotes. You will learn how to identify and interpret vertical asymptotes, horizontal asymptotes, and oblique or slant asymptotes and understand the conditions under which each type occurs. For a rational function �, we explain how denominator zeros can lead to vertical asymptotes, why common factors must first be simplified carefully, and how a removable discontinuity or “hole” can occur when both numerator and denominator become zero at the same value. You will also learn how the degrees of the numerator and denominator help determine horizontal asymptotes. When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is �; when the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients. We also explain when a rational function has an oblique asymptote, particularly when the numerator degree is exactly one greater than the denominator degree, and how polynomial long division can be used to determine the slant asymptote. The video presents a clear step-by-step procedure for sketching rational-function graphs: first identify and simplify the domain, then find x- and y-intercepts whenever they exist, determine vertical and horizontal or oblique asymptotes, analyze the behavior near excluded values and toward positive and negative infinity, and finally use all the information to construct an accurate graph. We work with examples such as �, showing how to determine the domain, x-intercept, y-intercept, vertical asymptote, horizontal asymptote, and overall graph behavior. We also examine examples where numerator and denominator share a common factor, demonstrating how simplification changes the visible graph while preserving the original domain restriction and producing a hole. Special attention is given to the relationship between algebraic expressions and graphical behavior so that students can understand not only how to calculate an answer but also why the graph behaves in a particular way. Important concepts covered include rational functions, polynomial numerator and denominator, domain restrictions, excluded values, zeros, roots, x-intercepts, y-intercepts, vertical asymptotes, horizontal asymptotes, oblique asymptotes, slant asymptotes, holes, removable discontinuities, end behavior, one-sided behavior. The examples and exercises are designed to help students develop mathematical reasoning, graph interpretation, algebraic manipulation, and problem-solving skills. Students should practice finding domains, identifying zeros, locating intercepts, determining asymptotes, simplifying rational expressions, analyzing behavior near excluded values, and sketching graphs accurately. This video is useful for Ethiopian University Freshman Mathematics, first-year university students, Grade 12 students preparing for university, mathematics learners, exam candidates, tutorial revision, homework practice, and anyone looking for a clear zero-to-hero explanation of rational functions. If you are studying Chapter 3: Functions, this Part 6 lesson provides a focused explanation of rational functions and their graphs and connects the algebraic form of a function with its graphical representation. 📚 Take notes, pause the video while working through the examples, practice the exercises independently, and review the graphing procedure until you can analyze a rational function step by step. Subscribe to Ephrata Academy Tube for more Ethiopian university freshman mathematics tutorials, chapter-by-chapter lessons, solved exercises, exam preparation, course reviews, academic revision, mathematics concepts, and zero-to-hero educational content. Share this lesson with classmates who are studying functions, rational functions, asymptotes, graph sketching, domain and range, intercepts, and university mathematics. 🔔 Keep learning, keep practicing, and build a strong mathematical foundation for university success.