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2.6 Rational Functions

2.6 Rational Functions . What is a Rational Function ?. * Also referred to as the reciprocal function The standard form of a rational function is: f(x ) = N(x) D(x) * A function is rational if ‘ x ’ is in the denominator, after the function has been simplified.

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2.6 Rational Functions

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  1. 2.6Rational Functions

  2. What is a Rational Function ? *Also referred to as the reciprocal function • The standard form of a rational function is: f(x)= N(x) D(x) *A function is rational if ‘x’ is in the denominator, after the function has been simplified

  3. Are the following functions rational?

  4. Finding Domain/Range of Rational Functions… • Domain will generally be all real numbers except for the Vertical Asymptote(s) or holes of the function • Range – generally all real numbers except for the Horizontal Asymptote or holes of the function

  5. Domain continued… • Ex: • -4 and -3 both make the denominator equal to zero, so they are both excluded from the domain • What is the domain of the following functions? • f(x)= • 2. f(x)=

  6. Describing the domain… Ex: f(x) = Description: As x decreases to 0 y increases without bound, as x increases to 0 y decreases without bound. Ex. 2: Describe the domain of f(x) =

  7. What are Asymptotes? • Asymptote- the line that the function approaches but never touches. • Holes - point of discontinuity (function is undefined at this value) Asymptotes

  8. Finding Vertical Asymptotes and Holes… • Vertical Asymptote- set the denominator equal to zero. • Holes – occur when a factor in the denominator is simplified (reduced to 1) by same factor in the numerator

  9. Identify any vertical asymptotes or holes in the following rational functions: • Ex 1: • Ex. 2:

  10. Horizontal Asymptotes…

  11. Cases of Horizontal Asymptotes • Case 1: N < D • Horizontal asymptote: y = k *most scenarios, y = 0 • Ex 1. • Ex 2. • Ex 3.

  12. Cases of Horizontal Asymptotes • Case 2: N = D • Horizontal asymptote is y = *the leading coefficients • Ex. 1) • Ex 2.)

  13. Cases of Horizontal Asymptotes • Case 3: N > D • Horizontal Asymptote is NONE *improper fraction! • This case will have a SLANT Asymptote • Ex. 1)

  14. Homework Day 1: • Pg 174 #’s 1, 7, 11, 13-16, 17, 21, 25, 33, 39 • Have fun     

  15. Day 2 – Warm Up • Without a calculator…. Please graph(not sketch) the rational function:

  16. What is a Slant Asymptote? • An oblique line that the graph approaches but never touches. • If the degree of the numerator is exactly 1 more than the denominator it has a slant asymptote

  17. Finding Slant Asymptotes • Check the Degree • Use either long or synthetic division to find asymptote • Find the Slant Asymptote • Ex 1.) • Ex 2.)

  18. Identify any asymptotes of the following rational functions… • f(x) = • f(x) = • f(x) =

  19. Graphing Rational Functions • Find the zeros of the denominator. These will be the vertical asymptotes/holes. Draw dotted line(s). • Find the horizontal asymptotes in the ways learned earlier. Draw dotted line(s). • See if there will be any slant asymptotes in the graph. Draw dotted line(s). • Find the zeros of the numerator. These will be the x-intercepts unless it is also a zero of the denominator. • Evaluate f(0) to find y intercept of function. Plot. • Create a table of values and plug in at least one point between and one point beyond each x-intercept and vertical asymptote. • Connect points with smooth curves

  20. Graphs of a Rational Function • Things to keep in mind: • Positive numerator, functions will be in quadrants 1 and 3 • Negative numerator, functions will be in quadrants 2 and 4 • Graph the following functions

  21. Some more practice with graphing… • 1) • 2) • 3)

  22. Homework Day 2… • Pg. 174-175 #’s 3, 9, 19, 23, 35, 45, 49, 51, 55, 69

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