Ri Review The Use of Graphing Calculators Solns

RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 16 Topic: Review the Use of Graphing Calculators In this revision, we will review the use of graphing calculators for the following topics. There may be tricks that you may not be aware of. 1: Graphing – Functions 2: Graphing – Parametric Equations 3: Equations & Inequalities 4: Complex Numbers 5: Discrete Random Variables 1: Graphing – Functions Skills Required: By using the FUNCTION mode, 1. Sketch the graphs of functions and composite functions [ o, p, q]. 2. Determine the y-value for a given x-value [/ value]. 3. Determine the x-intercepts [/ zero]. 4. Determine the coordinates of the minimum & maximum points [/ minimum/maximum] 5. Determine the points of intersection of 2 graphs [ / intersect]. 6. Determine the gradient of the function [ / d d y x OR nDeriv]. 7. Determine the area bounded by the function with the x -axis [/ ʃ f(x) dx OR fnInt]. 8. Sketch piecewise functions (OS 5.3.0 and 5.3.1) [» B: piecewise] Exercise 1 (a) Functions f and g are defined as follows: 2 21f( ) 2 xxx x −−= + , for ,x∈  2x≠− , ( )g( ) 3 e ln 1 xxx x −=−− − + , for 1x>− . Sketch, on separate diagrams, showing clearly any asymptotes, intersections with the axes and coordinates of stationary points, the graphs of (i) f( )yx= , (ii) g( )yx= [Note for Teachers: May need to guide students for (ii)]. The function h is defined as h: f ( ), for , 2.xx xx ∈ >−  (iii) Find the equation of the tangent to gh( )yx= at the point where x = 0. (iv) Find the area bounded by the graph of gh( )yx= with the x-axis.

(v) Solve the inequality gh( ) 2ln( 1)xx<+ . (b) Sketch the curve 1 101 , , 1 , 0 ,( 1) xy xx x xx −= + ∈ ≠− ≠+  stating clearly equations of any asymptotes, intersections with the axes and coordinates of turning points. (c) Sketch the graph of f( )yx= where 7 12 2 sin cos 1 for 0, f( ) ( 2)1 for 0 2.4 xx x x x x π+ − − ≤< = −−− ≤≤ State clearly the coordinates of the endpoints. Answers to Exercise 1(a) (i) Use 2 1 21 2 xxY x −−= + to sketch f( )yx= . Asymptotes: y = x – 4, x = –2 Intercepts: (–0.414, 0), (2.41, 0), (0, –0.5) Stationary Points: (0.646, –0.708), (–4.65, –11.3) (ii) Use ( )2 3 e ln 1xYx x −=−− − + to sketch g( )yx= . [Note for Teachers: GC will not show the complete graph below (regardless of zoom)] Asymptote: x = –1 Intercepts: (–0.999, 0), (4.76, 0), (0, –4) Stationary Point: (–0.659, –4.52) (iii) Use 3 21 ()Y YY= to sketch gh( )yx= Use [/ value] to obtain y = –4.46 at x = 0 OR Type 3Y (0) on the home screen. Use [/ d d y x ] to obtain d d y x = –0.487 at x = 0 OR Type 3 X0 d (Y )dX = on the home screen. Equation of tangent at x = 0 is y = – 0.487x – 4.46 (iv) Use [/ zero] to obtain x = –1.3044 and x =