Rational functions

Rational functions

• 1.
  RATIONAL FUNCTIONS A rational functionis a function of the form: ( ) ( ) ( )xq xp xR = where p and q are polynomials
• 2.
  ( ) () ( )xq xp xR = What would the domain of a rational function be? We’d need to make sure the denominator ≠ 0 ( ) x x xR + = 3 5 2 Find the domain.{ }3: −≠ℜ∈ xx ( ) ( )( )22 3 −+ − = xx x xH { }2,2: ≠−≠ℜ∈ xxx ( ) 45 1 2 ++ − = xx x xF If you can’t see it in your head, set the denominator = 0 and factor to find “illegal” values. ( )( ) 014 =++ xx { }1,4: −≠−≠ℜ∈ xxx
• 3.
  The graph oflooks like this:( ) 2 1 x xf = Since x ≠ 0, the graph approaches 0 but never crosses or touches 0. A vertical line drawn at x = 0 is called a vertical asymptote. It is a sketching aid to figure out the graph of a rational function. There will be a vertical asymptote at x values that make the denominator = 0 If you choose x values close to 0, the graph gets close to the asymptote, but never touches it.
• 4.
  Let’s consider thegraph ( ) x xf 1 = We recognize this function as the reciprocal function from our “library” of functions. Can you see the vertical asymptote? Let’s see why the graph looks like it does near 0 by putting in some numbers close to 0. 10 10 1 1 10 1 ==      f 100 100 1 1 100 1 ==      f 10 10 1 1 10 1 −= − =      −f 100 100 1 1 100 1 −= − =      −f The closer to 0 you get for x (from positive direction), the larger the function value will be Try some negatives
• 5.
  Does the functionhave an x intercept?( ) x xf 1 = There is NOT a value that you can plug in for x that would make the function = 0. The graph approaches but never crosses the horizontal line y = 0. This is called a horizontal asymptote. A graph will NEVER cross a vertical asymptote because the x value is “illegal” (would make the denominator 0) x 1 0 ≠ A graph may cross a horizontal asymptote near the middle of the graph but will approach it when you move to the far right or left
• 6.
  Graph ( ) x xQ 1 3+= Thisis just the reciprocal function transformed. We can trade the terms places to make it easier to see this. 3 1 += x vertical translation, moved up 3 ( ) x xf 1 = ( ) x xQ 1 3+= The vertical asymptote remains the same because in either function, x ≠ 0 The horizontal asymptote will move up 3 like the graph does.
• 7.
  Finding Asymptotes VERTICALASYMPTOTES There willbe a vertical asymptote at any “illegal” x value, so anywhere that would make the denominator = 0 ( ) 43 52 2 2 −− ++ = xx xx xR Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be. ( )( ) 014 =+− xx So there are vertical asymptotes at x = 4 and x = -1.
• 8.
  If the degreeof the numerator is less than the degree of the denominator, (remember degree is the highest power on any x term) the x axis is a horizontal asymptote. If the degree of the numerator is less than the degree of the denominator, the x axis is a horizontal asymptote. This is along the line y = 0. We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes. ( ) 43 52 2 +− + = xx x xR degree of bottom = 2 HORIZONTAL ASYMPTOTES degree of top = 1 1 1 < 2
• 9.
  If the degreeof the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at: y = leading coefficient of top leading coefficient of bottom degree of bottom = 2 HORIZONTAL ASYMPTOTES degree of top = 2 The leading coefficient is the number in front of the highest powered x term. horizontal asymptote at: 1 2= ( ) 43 542 2 2 +− ++ = xx xx xR 1 2 =y
• 10.
  ( ) 43 532 2 23 +− +−+ = xx xxx xR If thedegree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an oblique one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder. degree of bottom = 2 OBLIQUE ASYMPTOTES degree of top = 3 532 23 +−+ xxx432 −− xx remaindera5 ++x Oblique asymptote at y = x + 5
• 11.
  SUMMARY OF HOWTO FIND ASYMPTOTES Vertical Asymptotes are the values that are NOT in the domain. To find them, set the denominator = 0 and solve. To determine horizontal or oblique asymptotes, compare the degrees of the numerator and denominator. 1. If the degree of the top < the bottom, horizontal asymptote along the x axis (y = 0) 2. If the degree of the top = bottom, horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom 3. If the degree of the top > the bottom, oblique asymptote found by long division.
• 12.
  Acknowledgement I wish tothank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au