2024 JPJC Chapter 2 Graphing Techniques I

Chapter 2 Graphing Techniques I (Students’ version) / Pg 1 Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 References  http://www.h2maths.site [This website offers Applets which allow self-exploration of numerous types of graphs covered in this topic]  H2 Mathematics For 'A' Level, Federick Ho, David Khor, Yui-P'ng Lam, B.S. Ong Volume 1, Call No 510.76 HO  Mathematics The Core Course for A-level, L Bostock and Chandler, Call No 510 BOS  MEI Structured Mathematics (2 nd Edition), Pure Mathematics 4, Terry Heard and David Martin, Call No 510 HEA  Pure Mathematics 4, Hugh Neill and Douglas Quadling, Call No 510 NEI Prerequisites  Long division of polynomials (O level Add Math knowledge)  Graphs (including trigonometry)(O level Elementary and Add Math knowledge)  Quadratic graphs and their properties (O level Add Math knowledge)  Differentiation techniques and applications (O level Add Math knowledge)  Coordinate geometry and Further Coordinate Geometry (O level Add Math knowledge) Content requirement Include: • use of a graphic calculator to graph a given function • important characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes of the following: 2 2 2 2 1x y a b  2 2 2 2 2 2 2 2 1; 1x y y x a b a b    ax by cx d   2ax bx cy dx e    • determining the equations of asymptotes, axes of symmetry, and restrictions on the possible values of x and/or y • simple parametric equations and their graphs Chapter 2: Graphing Techniques I

Chapter 2 Graphing Techniques I (Students’ version) / Pg 2 Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 1 Basic Graphs 1.1 Straight Lines A straight line that is not a vertical line has an equation in the form y = mx + c, where m is the gradient of the line and c is the y-intercept. The equation in this form is called the gradient-intercept form. Lines that are not vertical Vertical line x y O c x y O c x y O c x y Ok 1.2 Quadratic Functions y = ax2 + bx + c, a  0 A quadratic function has an equation in the form 2y ax bx c   where 0a  . The graph of a quadratic function is a parabola and its concavity is determined by a. The graph has a turning point at 2 bx a  . Concavity of graph a > 0 Concave upwards xO y x = 2a b c xO x = 2a b c y xO y x = 2a b c a < 0 Concave downwards xO y x = 2a b c xO x = 2a b c y xO y x = 2a b c Note: If the equation of a quadratic graph is expressed in the form   2 y a x h k   , then the coordinates of the turning point are  ,h k .

Chapter 2 Graphing Techniques I (Students’ version) / Pg 3 Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Recall: Find the turning point of 22 5 3y x x   Solution: Method 1: 2 2 2 2 2 2 2 2 5 3 52 3 2 5 5 52 3 2 2 4 4 5 48 502 4 16 16 5 12 4 8 5 1The required turning point is , 4 8 y x x y x x y x x y x y x                                             Method 2: 2 2 2 5 3 d 4 5d dLet 0d 5Then 4 5 5 5 50 100 48 1when 2 5 34 4 4 16 8 y x x y xx y x x x y                          5 1The required turning point is , 4 8    

Chapter 2 Graphing Techniques I (Students’ version) / Pg 4 Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 1.3 Cubic Functions, y = ax3 + bx2 + cx + d, a  0 A cubic curve has an equation in the form y = ax3 + bx2 + cx + d , where a  0. A cubic curve intersects the x-axis at least once and at most thrice. Intersection(s) with x-axis once twice thrice a > 0 xO y d x y O d xO d y xO d y a < 0 xO y d x y O d xO d y xO d y 1.4 Exponential Functions Recall: The exponential constant, e = 2.71828… , is an irrational number. Note: For e , 0axy a  , the graph is as close as possible to, but never touches the x-axis while x approaches  . We say that the x-axis is the horizontal asymptote. Similarly, for e , 0axy a   , the graph is as close as possible to, but never touches the x-axis while x approaches  . We say that the x-axis is the horizontal asymptote. y x 1 O

Chapter 2 Graphing Techniques I (Students’ version) / Pg 5 Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 1.5 Logarithmic Functions . Recall: aa logyx x y   We denote elog x as ln x (natural logarithm) where 0x  . Note: For  ln , 0y ax a  , the graph is as close as possible to, but never touches the y-axis as x approaches 0 from the right. We say that the y-axis is the vertical asymptote. 1.6 Trigonometrical Functions (a) siny x The sine curve has the following features: (1) 1 1 y   (2) The sine curve has an amplitude equals to 1. (An amplitude is half the difference between the maximum and minimum values of y in a periodic function.) (3) The sine curve has a period of 2 radians. (Period is the interval the graph takes to go a complete cycle.) (b) cosy x The cosine curve has the following features: (1) 1 1 y   (2) The cosine curve has an amplitude equals to 1. (3) The cosine curve has a period of 2 radians. O y x  ln , 0y ax a  1 a

Chapter 2 Graphing Techniques I (Students’ version) / Pg 6 Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 (c) tany x The tangent curve has the following features: (1) y can take any real value. (2) As x approaches 3 5 2 2 2, , ,...,     the curve is closer and closer but never touches the lines 3 5 2 2 2, , ,...     x respectively. These lines are the vertical asymptotes. 1.7 The use of Graphing Calculator (GC) in drawing graphs Example 1 Sketch the following graphs with the help of a GC. (You may refer to Annex A) (a) 22 6y x x   (b) exy  (c) ( 2)( 4)y x x x   (d) lny x You should indicate the intersections with axes, turning points and asymptotes if possible. Solution: d) x y O c) x y O (Note: This graph requires zooming out or adjusting the window settings in order to see the whole graph) a) x y O b) x y O

Chapter 2 Graphing Techniques I (Students’ version) / Pg 7 Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Standard Window settings Keystrokes Screen shown on GC Explanation WINDOW Xmin: Extreme left value on x-axis Xmax: Extreme right value on x-axis Xscl: Distance between markings on the x-axis Ymin: Extreme bottom value on y-axis Ymax: Extreme top value on y-axis Yscl: Distance between markings on the y-axis Xres: Pixel resolution. You can always leave it as 1 Note: This setting is obtained by pressing ZOOM 6. However, if the graph plotted is not fully captured in this setting, you may change the values of Xmin, Xmax, Xscl, Ymin, Ymax and Yscl to capture the graph as clearly as possible. Lesson from Example 1 Despite the convenience of having the GC to assist us in curve sketching, it is not advisable to depend solely on the GC for doing so. As shown in Example 1, the shape of the curve that is shown on the GC may not always be correct due to Window settings or certain limitations in the GC. It is therefore still essential that we are familiar with the shape and characteristics of certain standard curves in order to sketch graphs quickly and accurately. More importantly, we must understand the concepts behind curve sketching and use the GC only a