Lesson 2.4 Exercises, pages

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1 Lesson. Eercises, pages 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

2 . Sketch the graph of each function. a) = - b) = 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: 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: 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

3 B 5. Sketch the graph of each function, then state the domain. a) b) = - 9 = 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: 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: 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

4 c) = d) = O ( )( ) 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) ( ) The quotient is 1; so the equation of the oblique asmptote is 1. asmptotes. Close to the vertical asmptote: When 0, 0.5 When 0, ( 1)( 1) 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

5 6. a) How are these functions different from other functions in this lesson? i) ii) = 3 - = Both functions contain 3 -terms. b) Sketch the graph of each function in part a, then state the domain and range ( 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

6 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 Join the points with smooth curves. The domain is: C 8. Sketch the graph of each function, then state the domain. a) = Use the factor theorem. Let f () Use mental math to determine f(1) 0 and f( 1) 0. f(3) 3 3 3(3) So, there are vertical asmptotes with equations: 1, 1, and 3 Since the degree of the numerator is less than the degree of the denominator, there is a horizontal asmptote with equation 0. Close to the asmptotes: 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

7 b) = Factor the numerator. Use the factor theorem. Let f() 3 Use mental math to determine f(1) 0 and f( 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 Since the degree of the numerator is 1 more than the degree of the denominator, there is an oblique asmptote. Determine: ( 1) ( ) 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: 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