PPT-Asymptotes

Jacques Paradis Professeur Département de mathématiques 2 Plan de la rencontre Élément de compétence Définition dasymptote Asymptotes verticales Asymptotes horizontales Levée dindéterminations. A horizontal asymptote is a sp cial case of a slant asymptote recipe for find ing a horizontal asymptote of a rational function Let deg Nx the degree of a numerator and deg Dx the degree of a denominator deg Nx deg Dx deg Nx deg Dx deg Nx deg Dx In particular if the degree of is strictly less than that of d then the axis will be the horizontal asymptotea geometrical condition that can be expressed analytically by saying 0 as and as If the degree of is greater than or equal to the degree of 1 The line is a vertical asymptote of the function if approaches as approaches from the right or left This graph has a vertical asymptote at 1 De64257nition 22 The line is a horizontal asymptote of the function if approaches as approaches This gra lim lim 87221 lim lim 87221 A function has a horizontal asymptote of provided lim 1 or lim 87221 If the function is a rational function a polynomial divided by a polynomial then we have some shortcuts for 57356nding asymptotes Shortcut for V Describe the end behavior of:. Graph the function. Determine the interval(s) on which the function is increasing and on which it is decreasing. . Lesson 3-7 Graphs of Rational Functions. Objective: 1. To graph rational functions.. Precalculus. : Do Now. Graph the function below by hand. Be sure to factor the function, find the horizontal and vertical asymptotes, and the x and y-intercepts . Only use a graphing calculator at the very end to confirm your answer. REMEMBER FAITS!!. AII.7 e . 2009. Objectives:. Find the Vertical Asymptotes. Find the Horizontal Asymptotes. Rational . Functions . A rational function can have more than one . vertical asymptote. , but it can have at most one . RevIEw. precalculus. y = (x+2). 2. y = (x-2). 2 . -1. y = -(x+1). 2 . + 3. 4) . The number of horsepower . H. required to overcome wind drag on a certain car is approximated by . H(s) = 0.002s. 2. + 0.05s - 0.029 , 0 < s < 100. Standard Form:. Transverse axis (axis that vertices lie on): Horizontal . Center (. h,k. ). Slopes of asymptotes: . a comes first!.  . Standard Form:. Transverse axis (axis that vertices lie on): Vertical . A bit more practice in Section 4.7b. Analysis of the Inverse Sine Function. 1. –1. D:. R:. Continuous. Increasing. Symmetry: Origin (odd . func. .). Bounded. Abs. Max. of at . x. = 1. Abs. Min. of at . A brief journey into Section . 4.5a. Analysis of the Tangent Function. by. Domain: All . reals. except odd. multiples of. Range:. Continuous on its domain. Increasing on each interval in. i. ts domain. . – . The Graph of a Rational . Function. 3 examples. General Steps to Graph a Rational Function. 1) Factor the numerator and the denominator. 2) State the domain and the location of any holes in the graph. Definition of a Vertical Asymptote. If f(x) approaches ±∞ as x approaches c from the left or right, then the line x = c is a . vertical asymptote. .. Vertical Asymptotes can be determined by finding where there is . ECEN2260 R. W. Erickson1.Bode plots: basic rulesA Bode plot is a plot of phase of a transfer function or otherdecibels, and phase in are plotted vs. frequency, using semi-logarithmic axes. The magni