Graphing Rational Functions and Asymptotes

Grade 10 Mathematics BC Curriculum - Functions and Relations 60-minute lesson

Learning Objectives

Define rational functions and recognize their general form Identify and classify vertical, horizontal, and oblique asymptotes Analyze behavior of rational functions near asymptotes Sketch basic graphs using intercepts, asymptotes, and end behavior Apply reasoning to solve rational function problems

What do you notice about this graph?

Look at the graph of f(x) = 1/x What happens as x approaches 0? What values can x NOT take? How does the graph behave at the edges?

What is a Rational Function?

A rational function has the form f(x) = p(x)/q(x) p(x) and q(x) are polynomials The denominator q(x) ≠ 0 Examples: f(x) = 1/x, f(x) = (2x+3)/(x-1) The domain excludes values where q(x) = 0

Vertical Asymptotes

Occur where the denominator equals zero: q(x) = 0 The function is undefined at these x-values Graph approaches but never touches these vertical lines Example: f(x) = (2x+3)/(x-1) has vertical asymptote at x = 1 Function 'blows up' near vertical asymptotes

Finding Horizontal Asymptotes

{"left":"Degree of numerator < degree of denominator\nHorizontal asymptote at y = 0\nDegrees are equal","right":"Horizontal asymptote at ratio of leading coefficients\nDegree of numerator > degree of denominator\nNo horizontal asymptote (possible oblique asymptote)"}

Oblique (Slant) Asymptotes

Occur when degree of numerator = degree of denominator + 1 Found using polynomial long division The quotient (ignoring remainder) gives the oblique asymptote Example: f(x) = (x² + 1)/(x - 2) Graph approaches this diagonal line as x → ±∞

Group Practice Activity

Groups of 5 students each Each group gets a different rational function Find vertical asymptotes Determine horizontal or oblique asymptotes Sketch rough graph on chart paper Show your calculations and reasoning

Graph Behavior Near Asymptotes

Independent Practice

Work individually on worksheet problems 5 problems of varying difficulty Find asymptotes algebraically Sketch graphs showing asymptotic behavior Interpret graphs to answer behavior questions Raise hand for help as needed

Exit Ticket Challenge

For f(x) = (x² + 1)/(x - 2) Find ONE vertical asymptote Find ONE horizontal OR oblique asymptote Write your answers on a sticky note Show your work briefly

Lesson Summary & Next Steps

Rational functions: f(x) = p(x)/q(x) where q(x) ≠ 0 Vertical asymptotes: where denominator = 0 Horizontal asymptotes: compare degrees of numerator and denominator Oblique asymptotes: when degree of numerator = degree of denominator + 1 Next: Transformations of rational functions and removable discontinuities