1. Further Mathematics Workshop Stowupland High School 8 th November 2005 See next slide for details of how the lesson on curve sketching was organised

2. Lesson followed the slides in this presentation Lesson followed the slides in this presentation “Role play” involved the students in a warm up exercise where they modelled, using their arms, the graphs “Role play” involved the students in a warm up exercise where they modelled, using their arms, the graphs Syllabus specification and key words were then covered with explanation Syllabus specification and key words were then covered with explanation Sketching the function (slide 7 ) was a teacher led explanation Sketching the function (slide 7 ) was a teacher led explanation Card matching activity used resource sheets that accompany this lesson Card matching activity used resource sheets that accompany this lesson Students were then given the handout with the opportunity to make their own notes against the six card matching graphs, explaining to themselves how the key features were identified Students were then given the handout with the opportunity to make their own notes against the six card matching graphs, explaining to themselves how the key features were identified Final activity was in pairs working on the questions at the end of the handout Final activity was in pairs working on the questions at the end of the handout

3. Role Play y = 0 y = 0 x = 0 x = 0 y = x y = x y = -x y = -x x = -y x = -y y = x 2 y = x 2 y = - x 2 y = - x 2 x = y 2 y = x 3 y = -x 3 y = sin x y = cos x y = tan x y = sin 2 x + cos 2 x

4. More role play…… G y = x G y = x M y = 2x M y = 2x G y = x 2 G y = x 2 M y = 2x 2 M y = 2x 2 G y = x 2 G y = x 2 M y = (x-1) 2 M y = (x-1) 2 G y = 1/x, x < 0 G y = 1/x, x < 0 M y = 1/x, x > 0 M y = 1/x, x > 0 G y = 1/x 2, x < 0 G y = 1/x 2, x < 0 M y = 1/x 2, x < 0 M y = 1/x 2, x < 0 this time in pairs! One person to be George and the other person Mildred. George to always stand in front of Mildred.

5. CURVE SKETCHING Treatment and sketching of graphs of rational functions. FP1C1 Be able to sketch the graph of y=f(x) obtaining information about symmetry, asymptotes parallel to the axes, intercepts with the co-ordinate axes, behaviour near x=0 and for numerically large x. Be able to ascertain the direction from which a curve approaches an asymptote. Be able to use a curve to solve an inequality.

6. Key words Rational function Rational function A function which can be expressed as N(x)/D(x) where N(x) and D(x) are both polynomials and D(x) is not the zero polynomial. A function which can be expressed as N(x)/D(x) where N(x) and D(x) are both polynomials and D(x) is not the zero polynomial. Polynomial Polynomial F(x) = a 0 +a 1 x+a 2 x 2 +a 3 x 3 +…..+a n x n F(x) = a 0 +a 1 x+a 2 x 2 +a 3 x 3 +…..+a n x n Asymptote Asymptote a straight line towards which a curve approaches but ever meet a straight line towards which a curve approaches but ever meet Sketch Sketch show axis intersections, asymptotes, and behaviour of the graph either side of any asymptote. show axis intersections, asymptotes, and behaviour of the graph either side of any asymptote.

7. Sketch the graph y = (3-x)/(2-x)(4-x) check where graph crosses axes. check where graph crosses axes. look for vertical asymptotes. look for vertical asymptotes. find behaviour as x approaches infinity. find behaviour as x approaches infinity. consider approach towards asymptotes. consider approach towards asymptotes. check with Autograph or graphical calculator check with Autograph or graphical calculator

8. Card matching activity match the six graphs with the six equations match the six graphs with the six equations Graph A…..Eqn R Graph B…..Eqn W Graph C…..Eqn P Graph D…..Eqn T Graph E…..Eqn Q Graph F…..Eqn S