Lesson. Eercises, pages 13 10 A 3. Sketch the graph of each function. ( - )( + 1) a) = b) = + 1 ( )( 1) 1 (- + )( - ) - ( )( ) 0 0 The function is undefined when: 1 There is a hole at 1. The function can be written as:, 1 The -coordinate of the hole is: 3 Draw an open circle at ( 1, 3). When 0, When 0, Draw the line on either side of the hole. There is a hole at. The function can be written as:, The -coordinate of the hole is: 0 Draw an open circle at (, 0). When 0, Draw the line on either side of the hole.. Sketching Graphs of Rational Functions Solutions DO NOT COPY. P
. Sketch the graph of each function. a) = - b) = - + + 1 1 0 0 6 The function is undefined when: 1 There are no common factors, so there are no holes. The vertical asmptote has equation: 1 There is a horizontal asmptote. The numerator and denominator have equal leading coefficients, so the horizontal asmptote has equation 1. Close to the asmptotes: 1.01 0.99 100 100 301 99 1.03 0.97 Some of the -values above are approimate. When 0, When 0, Determine the coordinates of some other points: (, ), (, ) asmptotes. Join the points to form smooth curves. 0 There are no common factors, so there are no holes. The vertical asmptote has equation: 0 There is a horizontal asmptote. The leading coefficients are and 1, so the horizontal asmptote has equation. Close to the asmptotes: 0.01 0.01 100 100 0 398.0 1.96 When 0, Determine the coordinates of some other points: (, 3), ( 1, 6), (1, ), (, 1) asmptotes. Join the points to form smooth curves. P DO NOT COPY.. Sketching Graphs of Rational Functions Solutions 3
B 5. Sketch the graph of each function, then state the domain. a) b) = - 9 = - - 9-9 9 0 0 The function is undefined when: 3 There are no common factors, so there are no holes. The vertical asmptotes have equations: 3 and 3 There is a horizontal asmptote with equation 0. Close to the asmptotes: 3.01.99.99 00 00 00 3.01 100 100 00 0.0 0.0 Some of the -values above are approimate. When 0, 0 Determine the approimate coordinates of some other points: (,.3), (, 1.6), (, 1.6), (,.3) asmptotes. Join the points to form smooth curves. The domain is: 3 0 There are no common factors, so there are no holes. The vertical asmptote has equation: 0 There is a horizontal asmptote. The leading coefficients are 1 and 1, so the equation of the horizontal asmptote is 1. Close to the asmptotes: 0.1 100 899 0.9991 When 0, 3 Determine the coordinates of some other points: (, 1.5) asmptotes. Join the points to form smooth curves. The domain is: 0. Sketching Graphs of Rational Functions Solutions DO NOT COPY. P
c) = + - 8 - - d) = - 3 + 1-8 6 O 3 1 1 8 ( )( ) Factor: ( ) There is a hole at. The function can be written as:, The -coordinate of the hole is: 6 Draw an open circle at (, 6), then draw the line on either side of the hole. The domain is: There are no common factors, so there are no holes. The vertical asmptote has equation: There is also an oblique asmptote. Determine: ( 3 1) ( ) 3 1 1 3 The quotient is 1; so the equation of the oblique asmptote is 1. asmptotes. Close to the vertical asmptote: 1.99.01 95 305 When 0, 0.5 When 0, 3 1 0 ( 1)( 1) 0 0.5 or 1 Plot points at (0, 0.5), (0.5, 0), and (1, 0). Determine the coordinates of some other points: ( 1, ), (3, 10), (, 10.5) Join the points to form smooth curves. The domain is: P DO NOT COPY.. Sketching Graphs of Rational Functions Solutions 5
6. a) How are these functions different from other functions in this lesson? i) ii) = 3 - = - 1 3 - - 1 Both functions contain 3 -terms. b) Sketch the graph of each function in part a, then state the domain and range. 1 3 3 1 0 ( 1)( 1) i) Factor: ( 1)( 1) There are holes at 1, and an asmptote with equation: 0 The function can be written as: 1, 1 The coordinates of the holes are: ( 1, 1) and (1, 1) Draw open circles at the holes. Graph 1 on either side of each hole. The domain is: 1, 0 The range is: 1, 0 ( 1)( 1) ii) Factor: ( 1)( 1) There are holes at 1. The function can be written as:, 1 The coordinates of the holes are: ( 1, 1) and (1, 1) Draw open circles at the holes. Graph on either side of the holes. The domain is: 1 The range is: 1 6. Sketching Graphs of Rational Functions Solutions DO NOT COPY. P
7. For a rational function, when the degree of the numerator is or more than the degree of the denominator, the graph has no horizontal or oblique asmptotes. Without using graphing technolog, determine a strateg to sketch the graph of then graph the function. State the domain. = 3 + The function is undefined when. Draw a vertical asmptote at. Make a table of values. Approimate the -values. 3 0 30 0 10 0 5 3.01 1.99 1 0 1 3 3 7 81 788 1 0 0.3 5. 11 Join the points with smooth curves. The domain is: C 8. Sketch the graph of each function, then state the domain. a) = 3-3 - + 3 3 3 3 0 Use the factor theorem. Let f () 3 3 3 Use mental math to determine f(1) 0 and f( 1) 0. f(3) 3 3 3(3) 3 3 0 So, there are vertical asmptotes with equations: 1, 1, and 3 Since the degree of the numerator is less than the 3 3 3 0 degree of the denominator, there is a horizontal asmptote with equation 0. Close to the asmptotes: 100 1.01 0.99 0.99 1.01.99 3.01 100 0.01 13 1 5 6 1.3 113 11 0.01 When 0, 0 Determine the approimate coordinates of other points: (, 0.3), ( 0.5, 0.1), (, 1.1) Draw smooth curves through the points. The domain is: 1, 1, 3 P DO NOT COPY.. Sketching Graphs of Rational Functions Solutions 7
b) = 3 - - + 3-0 0 Factor the numerator. Use the factor theorem. Let f() 3 Use mental math to determine f(1) 0 and f( 1) 0. 8 1 f() 3 () 0 ( 1)( 1)( ) The function is: ( )( ) There is a hole at. The function can be written as: ( 1)( 1), or 1, ( ), There is a vertical asmptote at. The -coordinate of the hole is 0.75. Since the degree of the numerator is 1 more than the degree of the denominator, there is an oblique asmptote. Determine: ( 1) ( ) 1 0 1 1 3 The quotient is ; so the equation of the oblique asmptote is. asmptotes. When 0, 0.5 When 0, 1 0, and 1 Choose points close to the asmptotes and other points: 3.01 1.99 7.5 8 30 96 Draw an open circle at (, 0.75). Join the points to form smooth curves. The domain is: 8. Sketching Graphs of Rational Functions Solutions DO NOT COPY. P