448 CHAPTER 6 Trigonometric Functions 4 Graph Functions of the Form csc ω( ) = + y A x B and sec ω( ) = + y A x B The role of A in these functions is to set the range. The range of = y x csc is { } ≤ − ≥ y y y 1or 1 ; the range of = y A x csc is { } ≤ − ≥ y y A y A or because of the vertical stretch of the graph by a factor of A. Just as with the sine and cosine functions, the period of ω( ) = y x csc and ω( ) = y x sec becomes π ω 2 because of the horizontal compression of the graph by a factor of ω 1 . The presence of B indicates that a vertical shift is required. Graphing a Function of the Form csc ω( ) = + y A x B Graph = − y x 2csc 1. Use the graph to determine the domain and the range of = − y x 2 csc 1. Solution EXAMPLE 3 We use transformations. Figure 75 shows the required steps. The domain of = − y x 2 csc 1 is π { } ≠ x x k k , aninteger . The range is { } ≤ − ≥ y y y 3 or 1 , or using interval notation, ( ] [ ) −∞ − ∪ ∞ , 3 1, . Check: Using a graphing utility, graph = − Y x 2 csc 1 1 to verify the graph shown in Figure 75(c). Figure 75 x y 1 x y 1 2 Multiply by 2; Vertical stretch by a factor of 2 x y 1 2 Subtract 1; Vertical shift down 1 unit ( , 2) ( , 1) ( , 1) (a) y 5 csc x (b) y 5 2 csc x (c) y 5 2 csc x 21 }{ }{ }{ 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5 2 5 2 5 2 5 5 5 5 5 5 5 5 5 Now Work PROBLEM 29 2. True or False If = x 3 is a vertical asymptote of the graph of a rational function R, then as ( ) → →∞ x R x 3, . (pp. 240–241) 1. The graph of = − − y x x 3 6 4 has a vertical asymptote. What is it? (pp. 240–241) ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 6.5 Assess Your Understanding 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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