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1 Oblique and Nonlinear Asymptote Activity Name: Prior Learning Reminder: Rational Functions In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form m n n1 n1 m1 m1 anx a x a0 m b x b x b 0 where an 0, b 0 and m and n are nonnegative integers. To make the notation a m little simpler, let s rename our rational function Nx ( ) f ( x) where Nx ( ) and are the numerator and denominator polynomials that define the rational function f ( x). The following chart provides an overview of how we determine vertical and horizontal asymptotes when dealing with rational functions. Nx ( ) Let f ( x) be a rational function, where Nx ( ) and have no common factors other than constants. The vertical asymptotes of f ( x) can be determined as follows: Finding vertical asymptotes Procedure Rational Function example Asymptotes of example Vertical asymptotes occur at each value of x which makes the denominator equal to 0. 15x 10 f ( x) 4 to 15x 10 f ( x) 4 simplifies 5(3x ) (3x)(3x) 5 3x Finding horizontal asymptotes Solving 3x 0yields x as a vertical 3 Note: 3x 0 does not yield a vertical asymptote, but does result in a hole in the graph at x. 3 Procedure Rational Function example Asymptote of example Whenever the degree of the numerator Nx ( ) is less than the degree of the denominator, the xaxis, which is the line 0, is a horizontal Whenever the degree of the numerator is equal to the degree of the denominator, the line a, where a and b are the b leading coefficients of the numerator and denominator, respectively, is the horizontal Whenever the degree of the numerator is greater than the degree of the denominator, there is no horizontal 15x 10 f ( x) has 1 as the 4 degree of the numerator and as the degree of the denominator. f ( x) has as the degree of the numerator and denominator. f ( x) 3 has as the degree of the numerator and 1 as the degree of the denominator. Since the degree of the numerator is less than the degree of the denominator, 0 is the horizontal Since the degrees of the numerator and denominator are the same and the leading coefficients of the numerator and denominator are 14 and 14 9, respectively, is the 9 horizontal Since the degree of the numerator is greater than the degree of the denominator, the function has no horizontal
2 Investigating Other Types of Asymptotes of Rational Functions Connection to Prior Learning: Rational Numbers Given r, use long division to find the quotient q and remainder r. Is equal to q 8 8 d that the divisor is the denominator. In this case, d = 8). Verify your answer., where d is the divisor? (Recall Extending to Rational Functions 1. Let s investigate a rational function where the degree of the numerator is one greater than the degree of the 3 denominator. For example, we can examine the rational function. a) Use either long division or synthetic division in the space provided to determine the quotient and remainder. (Hint: The remainder should be 45. If you don t get 45, raise your hand.) Quotient = Qx () Remainder = Rx ( ) 45 Numerator & Denominator of Rational Function Rational functions of the form Nx () Nx ( ) f ( x) can be written as Dx () ( ) ( ) ( ) N x R x y f x Q( x) D( x) D( x) where Qx ( ) and Rx ( ) are the quotient polynomial and the remainder polynomial, respectively, when we divide Nx ( ) by. Rewrite 3 ( ) in the form Q( x) Rx.
3 3 b) Let s examine the y values of the rational function and the y values of the quotient function y 16. To do this put the rational function in and the quotient function in. Next, set up your table by accessing and set the Indpnt variable to Ask so that you can input your x values into the table. Finally, select and enter the x values from the tables below, recording the corresponding y values. x x What can you say about the values in relative to the values in as x? 3 c) Graph the rational function and the quotient function y16 using your graphing calculator with the window settings below. Using two different colors, sketch the graphs of both in the box provided. x:[5,5] scale:5 y:[150,50] scale:5 d) Where do the graphs of the rational function and the quotient function look alike? Where do they look different?
4 3 e) Notice that as x the graph of approaches the graph of the quotient function y16. Since the quotient function s degree is one, its graph is a line. When the graph of a rational function approaches a nonhorizontal line as x, that line is called an oblique asymptote of the rational function. For our rational 3 function, y = 16 is an equation of the oblique Knowing that can be expressed as 45 3 y 16, explain why the graph of must approach the line y16 as x. 45 (Hint: Think about what happens to as x.) f) In part e above you explained why the graphs are close to each other as x. Give one possible reason why the graphs are not close to each other near x 3? 3 g) In part e above you learned that the rational function has an oblique asymptote that was obtained by dividing the numerator of the rational function by the denominator of the rational function. Also, you learned that an oblique asymptote of a rational function is a nonhorizontal line, so it is defined by a degree 1 polynomial. What must the relationship between the degrees of the numerator and the denominator of a rational function be in order for the rational function to have an oblique asymptote? Explain.
5 3 x 7x 13x 18. Now let s consider the rational function where the degree of the numerator is greater than x 1 the degree of the denominator. a) Use long division or synthetic division to determine the quotient polynomial Qx () and the remainder polynomial Rx (). Then write the function in the form of ( ) Q( x) Rx. Qx () Rx ( ) Rx () Q() x Dx () How does the degree of this quotient polynomial differ from the degree of the quotient polynomial in problem 1 a)? What could be the reason for this difference? b) As in 1b) examine the y values of the rational function and the quotient function Q( x ). Remember to let represent the rational function and the quotient function. x x What can you say about the values of the rational function relative to the values of the quotient function as x?
6 c) Graph the rational function and the quotient function using your graphing calculator with the settings below. Using two different colors, sketch the graphs of both in the box provided. x:[10,10] scale:1 y:[40,100] scale:0 3 x 7x 13x 18 d) Notice that as x the graph of approaches the graph of the quotient function x 1 Q( x ). This quotient function is a nonlinear asymptote of our rational function. This asymptote is nonlinear 3 x 7x 13x 18 because the quotient polynomial s degree is not 1. Knowing that can be expressed as x 1 3 x 7x 13x 18 Q( x), explain why the graph of must approach the graph of Q( x) as x 1 x 1 x. (Hint: If needed, refer to the hint in 1e.) 3. a) Given the rational function 5 3 x 5x x x1 find the following: Qx () R( x) 10x 40 Rx () Q() x Dx () The degree of the numerator is how much greater than the degree of the denominator? What is the degree of the quotient polynomial? Your answers to the previous two questions should be the same. Why should they be the same?
7 b) Graph the rational function and the quotient function using your graphing calculator with the settings below. Sketch the graphs of both in the box provided. x:[8,8] scale:1 y:[500,500] scale:100 c) What do the graphs in part b) suggest to you about the rational function and its corresponding quotient function? Explain. 4. Given the rational function 5 3 x 58x 00x x x 56x 56x 144. x1 x1 a) Use the above to determine the asymptote of the graph of (Note that the division has already been done for you) 5 3 x 58x 00x x 1 as x. b) Verify your answer by graphing the rational function and your asymptote in the box provided. x:[6,6] scale:1 y:[00,400] scale:50 5. Find the equation of the asymptote as x for each of the following rational functions. Graph each rational function and asymptote in a suitable window to verify your results. a) 4 3 x x x x x b) x x x c) 3 x x x 4 9 x 3
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