Rational Functions
- 1. RATIONAL FUNCTIONS A rational function is 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 of looks 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 the graph ( ) 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 function have 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+= This is 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 will be 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 degree of 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 degree of 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 the degree 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 HOW TO 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. Problem 1 • In an inter-barangay basketball league, the team from Barangay Culiat has won 12 out of 25 games, a winning percentage of 48%. How many games should they win in a row to improve their win percentage to 60%?
- 13. From Problem No. 1 • We have seen that they need to win 8 games consecutively to raise their percentage to atleast 60%. What will be their winning percentage if they win (a) 10 games in a row (b) 15? 20? 30? 50? 100 games?
- 14. Why Should You Learn This? • Rational functions are used to model and solve many problems in the business world. • Some examples of real-world scenarios are: – Average speed over a distance (traffic engineers) – Concentration of a mixture (chemist) – Average sales over time (sales manager) – Average costs over time (CFO’s)
- 15. Domain Find the domain of 2x 1f(x) + = Denominator can’t equal 0 (it is undefined there) 2 0 2 x x + ≠ ≠ − ( ) ( )Domain , 2 2,−∞ − ∪ − ∞ Think: what numbers can I put in for x????
- 16. You Do: Domain Find the domain of 2)1)(x(x 1-xf(x) ++ = Denominator can’t equal 0 ( ) ( )1 2 0 1, 2 x x x + + ≠ ≠ − − ( ) ( ) ( )Domain , 2 2, 1 1,−∞ − ∪ − − ∪ − ∞
- 17. You Do: Domain Find the domain of = +2 xf(x) x 1 Denominator can’t equal 0 2 2 1 0 1 x x + ≠ ≠ − ( )Domain ,−∞ ∞
- 18. Vertical Asymptotes At the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the problems below. 2x 1f(x) + = 2x =− 2)1)(x(x 1-xf(x) ++ = 1, 2x x= − = − = +2 xf(x) x 1 none
- 19. Vertical Asymptotes The figure below shows the graph of 2x 1f(x) + = The equation of the vertical asymptote is 2x=−
- 20. Vertical Asymptotes • Set denominator = 0; solve for x • Substitute x-values into numerator. The values for which the numerator ≠ 0 are the vertical asymptotes
- 21. Example • What is the domain? x ≠ 2 so What is the vertical asymptote? x = 2 (Set denominator = 0, plug back into numerator, if it ≠ 0, then it’s a vertical asymptote) ( ,2) (2, )−∞ ∪ ∞ 2 2 3 1 ( ) 2 x x f x x − − = −
- 22. You Do Domain: x2 + x – 2 = 0 (x + 2)(x - 1) = 0, so x ≠ -2, 1 Vertical Asymptote: x2 + x – 2 = 0 (x + 2)(x - 1) = 0 Neither makes the numerator = 0, so x = -2, x = 1 ( , 2) ( 2,1) (1, )−∞ − ∪ − ∪ ∞ 2 2 2 7 4 ( ) 2 x x f x x x + − = + −
- 23. The graph of a rational function NEVER crosses a vertical asymptote. Why? • Look at the last example: Since the domain is , and the vertical asymptotes are x = 2, -1, that means that if the function crosses the vertical asymptote, then for some y-value, x would have to equal 2 or -1, which would make the denominator = 0! ( , 1) ( 1,2) (2, )−∞ − ∪ − ∪ ∞ 2 2 2 7 4 ( ) 2 x x f x x x + − = + −
- 24. Examples f x x ( ) = + 4 12 f x x x ( ) = + 2 3 12 What similarities do you see between problems? The degree of the denominator is larger than the degree of the numerator. Horizontal Asymptote at y = 0 Horizontal Asymptote at y = 0
- 25. Examples h x x x ( ) = + + 2 1 1 82x 15x g(x) 2 2 − + = What similarities do you see between problems? The degree of the numerator is the same as the degree or the denominator. Horizontal Asymptote at y = 2 Horizontal Asymptote at 5 2 y =
- 26. Examples 13x 54x5x3x f(x) 23 + −+− = 2x 9x g(x) 2 + − = What similarities do you see between problems? The degree of the numerator is larger than the degree of the denominator. No Horizontal Asymptote No Horizontal Asymptote
- 27. Asymptotes: Summary 1. The graph of f has vertical asymptotes at the _________ of q(x). 2. The graph of f has at most one horizontal asymptote, as follows: a) If n < d, then the ____________ is a horizontal asymptote. b) If n = d, then the line ____________ is a horizontal asymptote (leading coef. over leading coef.) c) If n > d, then the graph of f has ______ horizontal asymptote. zeros line y = 0 no a y b =
- 28. You Do Find all vertical and horizontal asymptotes of the following function ( ) 2 1 1 x f x x + = + Vertical Asymptote: x = -1 Horizontal Asymptote: y = 2
- 29. You Do Again Find all vertical and horizontal asymptotes of the following function ( ) 2 4 1 f x x = + Vertical Asymptote: none Horizontal Asymptote: y = 0
- 30. Oblique/Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator. Long division is used to find slant asymptotes. The only time you have an oblique asymptote is when there is no horizontal asymptote. You cannot have both. When doing long division, we do not care about the remainder.
- 31. Example Find all asymptotes. ( ) 2 2 1 x x f x x − − = − Vertical x = 1 Horizontal none Slant ( ) 2 2 1 2 -2 x x x x x x − − − − − y = x
- 32. Example • Find all asymptotes: 2 2 ( ) 1 x f x x − = − Vertical asymptote at x = 1 n > d by exactly one, so no horizontal asymptote, but there is an oblique asymptote. ( ) 2 2 1 1 2 2 ( 1) 1 - x x x x x x x + − − − − − − − y = x + 1