PPT-Sec 5: Vertical Asymptotes & the Intermediate Value Theorem

Definition of a Vertical Asymptote If fx 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 . Then there exists a number in ab such that The idea behind the Intermediate Value Theorem is When we have two points af and bf connected by a continuous curve The curve is the function which is Continuous on the interval ab and is a numb 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.. Continuity. A function is . continuous. if you can draw the graph without picking up your pencil.. Definition:. A function . y = f(x). is continuous at an interior point . c. in its domain if. (the limit has to equal the value of the function at . 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 . 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 . Obj. : Understand and use vertical angle theorem. Why do we need Proofs??????. 1 region. 1 2. 2 regions. 4 regions. 8 regions. 16 regions. How many regions will be in a circle with 6 pts. ?????. 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. Today’s Topics. Academic History. Teaching and Research Philosophies. Current and Future Research. Service & Grants. Existence and Generation of Hamiltonian cycles in the Cartesian product of two directed cycles after a .  . Lesson Aim: . How do we prove and apply theorems about . angles. ?.  . Lesson Objectives:. SWBAT . To prove and apply theorems about angle.. NYS Content Strand.  . G.PS.4. Construct various types of reasoning, arguments, justifications and methods of proof for problems.. 4. 3. 2. 1. 0. In addition to level 3.0 and beyond what was taught in class, the student may. : . Make connection with other concepts in math.. Make connection with other content areas.. Explain the relationship between the Pythagorean Theorem and the distance formula.. Writing Rational Functions Honors Algebra II Keeper Think Backwards!!! Example: Write a rational function f that has a vertical asymptote at , a horizontal asymptote and a zero at .   Example: Write a rational function g with vertical asymptotes at