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Rational Functions and Their Graphs

Rational Functions and Their Graphs

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Rational Functions and Their Graphs

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  1. Rational Functions and Their Graphs Section 2.6 Page 326

  2. Definitions • Rational Function- a quotient of two polynomial functions in the form f(x) = p(x) q(x) ≠ 0 q(x) • Domain:

  3. Example 1 • Find the domain of each rational function

  4. Reciprocal Function Arrow Notation (see page 328)

  5. Arrow Notation

  6. Use the graph to answer the following questions. • As x → -2-, f(x) → • As x → -2+, f(x) → • As x → 2-, f(x) → • As x → 2+, f(x) → • As x → -, f(x) → • As x → , f(x) →

  7. Vertical Asymptotes • Definition: the line x = a is a vertical asymptote of the graph of a function if f(x) increases or decreases (goes to infinity) without bound as x approaches a • Locating Vertical Asymptotes: set the denominator of your rational function equal to zero and solve for x Find the vertical asymptotes of f(x) = x – 1 x2 – 4

  8. Homework • Page 342 #1 - 28

  9. Holes • A value where the denominator of a rational function is equal to zero does not necessarily result in a vertical asymptote. • If the numerator and the denominator of the rational function has a common factor (x – c) then the graph will have a hole at x = c • Example: f(x) = (x2 – 4) x – 2

  10. Finding the Horizontal Asymptote First identify the degree (highest power) of p(x) and q(x). f(x) = p(x) degree n q(x) degree m and identify their leading coefficients.

  11. Find the Vertical and Horizontal Asymptotes

  12. Review Transformation of Functions • Describe how the graphs of the following functions are transformed from its parent function.

  13. Homework • Page 342 #29 - 48

  14. Graphing Rational Functions • Seven Step Strategy – page 334 • Check for symmetry • Find the intercepts • Find the asymptotes – check for holes • Plot additional points as necessary

  15. Example 6 – Graph • Symmetry • Intercepts • Asymptotes • Plot points

  16. Example – Graph Symmetry Intercepts Asymptotes Plot points

  17. Slant Asymptotes • Slant Asymptotes occur when the degree of the numerator of a rational function is exactly one greater than that of the denominator • Note- when the degrees are the same or the denominator has a greater degree the function has a horizontal asymptote. Line l is a slant asymptote for a function f(x) if the graph of y = f(x) approaches l as x → ∞ or as x → -∞ l

  18. Determine the Slant Asymptote • Use synthetic division to find the slant asymptote then graph the function

  19. Find the Slant Asymptote use long division

  20. Partner Work Check for symmetry then find the intercepts, asymptotes, and holes of each rational function

  21. Homework • Page 342 #49 – 78 do 2 skip 1