1. MCB4U - Santowski1 C7 – Asymptotes of Rational and Other Functions IB Math HL/SL - Santowski

2. MCB4U - Santowski2 (A) Introduction To help make sense of any of the following discussions, graph all equations and view the resultant graphs as we discuss the concepts It may also be helpful to use the graphing technology to generate a table of values as you view the graphs Use either WINPLOT or a GDC

3. MCB4U - Santowski3 (A) Review of Rational Functions A rational number is a number that can be written in the form of a fraction. So likewise, a rational function is function that is presented in the form of a fraction. We have seen two examples of rational functions in this course  we can generate a graph of polynomials when we divide them in our work with the Factor Theorem  i.e. Q(x) = (x 3 - 2x + 1)/(x - 1). If x-1 was a factor of P(x), then we observed a hole in the graph of Q(x). If x-1 was not a factor of P(x), then we observed an asymptote in the graph of Q(x)

4. MCB4U - Santowski4 (A) Review of Rational Functions

5. MCB4U - Santowski5 (A) Review of Rational Functions We have seen several examples of rational functions in this course, when we investigated the reciprocal functions of linear fcns, i.e. f(x) = 1/(x + 2) and quadratic fcns, i.e. g(x) = 1/(x 2 - 3x - 10) and the tangent function y = tan(x).

6. MCB4U - Santowski6 (B) Domain, Range, and Zeroes of Rational Functions Given the rational function r(x) = n(x)/d(x), The domain of rational functions involve the fact that we cannot divide by zero. Therefore, any value of x that creates a zero denominator is a domain restriction. Thus in r(x), d(x) cannot equal zero. For the zeroes of a rational function, we simply consider where the numerator is zero (i.e. 0/d(x) = 0). So we try to find out where n(x) = 0 To find the range, we must look at the various sections of a rational function graph and look for max/min values EXAMPLES: Graph and find the domain, range, zeroes of f(x) = 7/(x + 2), g(x) = x/(x 2 - 3x - 4), and h(x) = (2x 2 + x - 3)/(x 2 - 4)

7. MCB4U - Santowski7 (B) Domain, Range, and Zeroes of Rational Functions EXAMPLES: Graph and find the domain, range, zeroes of f(x) = 7/(x + 2), g(x) = x/(x 2 - 3x - 4), and h(x) = (2x 2 + x - 3)/(x 2 - 4)

8. MCB4U - Santowski8 (C) Vertical and Horizontal Asymptotes Illustrate with a graph of y = 1/x and draw several others (i.e. pg 348) A vertical asymptote occurs when the value of the function increases or decreases without bound as the value of x approaches a from the right and from the left. We symbolically present this as f(x)  + ∞ as x  a + or x  a - We re-express this idea in limit notation  lim x  a+ f(x) = + ∞ A horizontal asymptotes occurs when a value of the function approaches a number, L, as x increases or decreases without bound. We symbolically present this as as f(x)  + a as x  + ∞ or x  - ∞ We can re-express this idea in limit notation  lim x  ∞ f(x) = a

9. MCB4U - Santowski9 (D) Finding the Equations of the Asymptotes To find the equation of the vertical asymptotes, we simply find the restrictions in the denominator and there is our equation of the asymptote i.e. x = a To find the equation of the horizontal asymptotes, we can work through it in two manners. First, we can prepare a table of values and make the x value larger and larger positively and negatively and see what function value is being approached The second approach, is to rearrange the equation to make it more obvious as to what happens when x gets infinitely positively and negatively.

10. MCB4U - Santowski10 (D) Finding the Equations of the Asymptotes ex. Find the asymptotes of y = (x+2)/(3x-2) So we take lim x  2/3+ f(x) = + ∞ and lim x  2/3- f(x) = - ∞  thus we have an asymptote at x = 2/3 To find the horizontal asymptote  a table of values (or simply large values for x) returns the following values: x = 10 9  f(10 9 ) = 0.3333333342 or close to 1/3 x = -(10 9 )  f(-(10 9 )) = 0.3333333342 or close to 1/3

11. MCB4U - Santowski11 (D) Finding the Equations of the Asymptotes Alternatively, we can find the horizontal asymptotes of y = (x+2)/(3x-2) using algebraic methods  divide through by the x term with the highest degree as x  + ∞, then 2/x  0

12. MCB4U - Santowski12 (E) Examples Further examples to do  Find vertical and horizontal asymptotes for: y = (4x)/(x 2 +1) y = (2-3x 2 )/(1-x 2 ) y = (x 2 - 3)/(x+5)

13. MCB4U - Santowski13 (E) Examples y = (4x)/(x 2 +1) y = (2-3x 2 )/(1-x 2 ) y = (x 2 - 3)/(x+5)

14. MCB4U - Santowski14 (F) Graphing Rational Functions If we want to graph rational functions (without graphing technology), we must find out some critical information about the rational function. If we could find the asymptotes, the domain and the intercepts, we could get a sketch of the graph ex => f(x) = (x 2 )/(x 3 -2x 2 - 5x + 6) NOTE: after finding the asymptotes (at x = -2, 1,3) we find the behaviour of the fcn on the left and the right of these asymptotes by considering the sign of the ∞ of f(x).

15. MCB4U - Santowski15 (F) Oblique Asymptotes Some asymptotes that are neither vertical or horizontal => they are slanted. These slanted asymptotes are called oblique asymptotes. Ex. Graph the function f(x) = (x 2 - x - 6)/(x - 2) (which brings us back to our previous work on the Factor Theorem and polynomial division) Recall, that we can do the division and rewrite f(x) = (x 2 - x - 6)/(x - 2) as f(x) = x + 1 - 4/(x - 2). Again, all we have done is a simple algebraic manipulation to present the original equation in another form. So now, as x becomes infinitely large (positive or negative), the term 4/(x - 2) becomes negligible i.e. = 0. So we are left with the expression y = x + 1 as the equation of the oblique asymptote.

16. MCB4U - Santowski16 (G) Internet Links Rational Functions from WTAMU Calculus@UTK 2.5 - Limits Involving Infinity Calculus@UTK 2.5 - Limits Involving Infinity Calculus I (Math 2413) - Limits - Limits Involving Infinity from Paul Dawkins Calculus I (Math 2413) - Limits - Limits Involving Infinity from Paul Dawkins Limits Involving Infinity from P.K. Ving

17. MCB4U - Santowski17 (G) Homework MCB4U: DAY 1; Nelson text, p356, Q1-4 DAY 2; Nelson text, p357, Q10,11,12,14,15 IB Math HL/SL: Stewart, 1989, Chap 5.1, p212, Q2,3 Stewart, 1989, Chap 5.2, p222, Q2-6 Stewart, 1989, Chap 5.6, p244, Q1,2