Chapter. Part 1: Consider the function
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1 Chapter Analysing rational Functions Pages Part 1: Consider the function a) What value of x is important to consider when analysing this function? b) Now look at the graph of this function below. 1
2 Notice there is a point of discontinuity at (2, 3). point of discontinuity a point, described by an ordered pair, at which the graph of a function is not continuous. How can you simplify the function? ***NOTE: When you cancel the like factors from the top and the bottom, that x value can give you the point of discontinuity, and the intercepts. 2
3 Determine the x and y intercept for the function 3
4 Part 2: Consider the function What value of x is important to consider when analysing this function? Now look at the graph of this function below. 4
5 What is happening at x = 2? How can you simplify the function? *** NOTE: Because you cannot cancel any like factors from the top and the bottom, the nonpermissible value found from the bottom gives you a vertical asymptote without having to graph. 5
6 Determine the x and y intercept for the function 6
7 Example 2 Page 448 Rational Functions: Points of Discontinuity Versus Asymptotes a) Compare the behaviour of the functions and near any non permissible values. b) Explain any differences. 7
8 Example 2: Your Turn Page 449 Compare the functions and explain any differences. and Answer 8
9 Determining Asymptotes for Rational Functions: #1 Vertical Asymptotes occur at the value(s) of x which make(s) the denominator of the simplified expression equal 0. Example: Determine the vertical asymptote(s) for 9
10 #2 Horizontal Asymptotes occur when the degree of the numerator the degree of the denominator. Case #1: d of n < d of d => y = 0 Case #2: d of n = d of d => y = division of leading coefficients Example: Determine the horizontal asymptote for the following: 10
11 #3 Oblique(slant) Asymptotes occur when the degree of the numerator is more than the degree of the denominator. Example: Determine the oblique asymptote for the following: 11
12 Examples: For each of the following rational functions, predict the locations of any asymptotes, points of discontinuity, and intercepts. 12
13 Example 3 Page 449 Match Graphs and Equations for Rational Functions Match the equation of each rational function with the most appropriate graph. For each of the rational functions, predict the locations of any asymptotes, points of discontinuity, and intercepts. 13
14 Example 3: Your Turn Page 450 Match the equation of each rational function with the most appropriate graph. For each of the rational functions, predict the locations of any asymptotes, points of discontinuity, and intercepts. Answer 14
15 More Examples: Write the equation of a possible rational function with each set of characteristics. a) vertical asymptote of x = 2, a point of discontinuity at x intercept of 0 and y intercept of 0. b) vertical asymptote of x = 3, no point of discontinuity, x intercepts of 1 and 6, and y intercept of 2. c) no vertical asymptote, point of discontinuity at x intercept of 2 and y intercept of 2 15
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INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC Surds Page 1 Algebra of Polynomial Functions Page 2 Polynomial Expressions Page 2 Expanding Expressions Page 3 Factorising Expressions
Section 4.3 Concavity and Curve Sketching 1.5 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.3 Concavity and Curve Sketching 1.5 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) Concavity 1 / 29 Concavity Increasing Function has three cases (University of Bahrain)
Curve Sketching. The process of curve sketching can be performed in the following steps:
Curve Sketching So ar you have learned how to ind st and nd derivatives o unctions and use these derivatives to determine where a unction is:. Increasing/decreasing. Relative extrema 3. Concavity 4. Points
( ) c. m = 0, 1 2, 3 4
G Linear Functions Probably the most important concept from precalculus that is required for differential calculus is that of linear functions The formulas you need to know backwards and forwards are: