Lecture Notes

SAJC 2020 JC1 H2 Mathematics Lecture Notes Chapter 4: Graphing Techniques Page 1 of 39 Chapter 4: Graphing Techniques Table of Content 1 Basic Features of Graphs 1.1 Asymptotes (A) (a) Vertical Asymptotes (b) Horizontal Asymptotes (c) Oblique Asymptotes 1.2 Axial Intercepts (I) 1.3 Stationary points (S) 1.4 Symmetries 2. Graphs of Rational Functions 2.1 Degree of P(x) < Degree of Q(x) 2.2 Degree of P(x) = Degree of Q(x) 2.3 Degree of P(x) > Degree of Q(x) 3. Graphs of Conic Sections 3.1 Parabola 3.2 Circles 3.3 Ellipse 3.4 Hyperbolas 3.5 Conics App in GC 4. Parametric Equations 4.1 Graphs 4.2 Converting Parametric equations to Cartesian Equations 5 Modelling the Projectile Motion using Parametric Equations 6. Self-reading Examples Annex 1 GC Keystrokes for finding Maximum Point and Axial Intercepts.

SAJC 2020 JC1 H2 Mathematics Lecture Notes Chapter 4: Graphing Techniques Page 2 of 39 Prerequisite Knowledge: You should be able to  Perform long division,  Able to complete the square for quadratic expressions,  Apply differentiation techniques to find stationary points,  Applying second derivative test to find nature of stationary points,  Sketch the graphs of basic graphs such as linear function ()y mx c , quadratic function 2()y ax bx c   and their properties such as estimation of gradient, maximum and minimum points and symmetry,  Sketch the graphs of exponential function   xy ka and logarithmic function  logayx where a is a positive integer. Objectives: At the end of the chapter, you should be able to  Use Graphing Calculator to graph a given function  Understand important characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes of the following: 22 22 1xy ab 22 22 1xy ab ; 22 22 1yx ba ax by cx d   2ax bx cy dx e    Determine the equations of asymptotes, axes of symmetry and restrictions on the possible values of x and/or y  Understand and draw simple parametric equations and their graphs.

SAJC 2020 JC1 H2 Mathematics Lecture Notes Chapter 4: Graphing Techniques Page 3 of 39 1 Basic Features of Graphs Purpose of a Graph A graph illustrates diagrammatically the relationship between two variables, usually denoted as x and y. This helps us to ‘visualise’ the relationship between x and y which can aid us in understanding the relationship between x and y. In secondary mathematics, we learnt to plot the graph of a function. It is done by plotting points on a graph paper. Unlike plotting a graph, in curve sketching we are not required to find all points of the function. Instead, we are required to show important features of the graph, such as  Asymptotes (A)  Intersection with axes (I)  Stationary points (S)  Symmetry In particular, the first three are the more commonly seen and important basic features of a graph and you can remember them using the acronym “A.I.S”. 1.1 Asymptotes (A) In our syllabus, we will learn three types of asymptotes, namely, vertical asymptote, horizontal asymptote and oblique asymptote. Asymptotes are usually drawn as dotted lines. (a) Vertical Asymptote In the graph of y = f(x), if there exist a constant a such that ,x a y  or as ,x a y  , then the line x = a is a vertical asymptote of the graph. For example, in the graph of ln , 0y x x , the value of y tends to  when x gets closer and closer to 0. (In fact, x approaches to 0 from the right side, i.e. x  0+). Then the line ___________ is a vertical asymptote of the graph y = ln x. Notation: If x approaches a value a from the right-hand side but never reaching a, we write xa  . Similarly, if x approaches a value a from the left-hand side but never reaching a, we write xa  . y x O x = 0 1 Learning points  Have you ever wondered why y = ln x does not touch the y – axis?

SAJC 2020 JC1 H2 Mathematics Lecture Notes Chapter 4: Graphing Techniques Page 4 of 39 Note: 1. A graph does not touch its vertical asymptote(s) at all times. 2. GC does not indicate the presence of asymptotes. For example, the graph y = ln x , obtained from GC is shown, appears as if the graph discontinues approximately at the point (0.15, – 2)! This is due to the limitation of GC. Example 1 Find the equation of vertical asymptotes (if any) of the following graphs with equation (a) 2 1y x  (b) 3 2 5 xy x  (c) ln (1 2 )yx Solution: (b) Horizontal Asymptote In the graph of y = f(x), if there exist a constant k such that ,x y k  or as ,x y k  , then the line y = k is a horizontal asymptote of the graph. For example, for the graph of exy , As ,e xxy   , but , e 0 xxy   . The line ___________ is a horizontal asymptote. x y O 1 y = 0 Learning points  NOTE: DO NOT copy the graph from the GC blindly!

SAJC 2020 JC1 H2 Mathematics Lecture Notes Chapter 4: Graphing Techniques Page 5 of 39 Remark: A graph may cut across a horizontal asymptote. However, positive or negative infinity, the graph does not touch the horizontal asymptote. (c) Oblique Asymptote If the graph of y = f(x) approaches the line y = mx + c (where m  0) as x and x , then the line y = mx + c is an oblique asymptote for the curve y = f(x). Consider the graph of 2 12yx x   , as the x-value gets larger and larger, the y-value seems to get closer and closer to a certain non- horizontal, non-vertical straight line. As x , As x , Therefore we say that _________________is an oblique asymptote. y y = x + 2 2 x -2 O Learning points  Commented [YWJ1]: Changed from and to or

SAJC 2020 JC1 H2 Mathematics Lecture Notes Chapter 4: Graphing Techniques Page 6 of 39 1.2 Axial Intercepts (x and y intercepts) (I) These are the point where the curve intersects x- and y-axes. Obtain the x- and y-intercepts by setting y = 0 and x = 0 respectively. Example 2 Find the axial intercepts of 2e1xyx . Solution: y-intercept: When x = 0, 1y  . x-intercept: Hence, the axial intercepts are Self-learning activity: See ANNEX 1- AXIAL INTERCEPTS (pg 35) to find out how to use GC to find x- and y-intercepts on a graph. 1.3 Stationary Points (S) If d d y x = 0 when x = a, then the stationary points occurs at x = a. To determine the nature of the stationary points we can use the second derivative test. Maximum point Minimum point 2 2 d 0d xa y x   2 2 d 0d xa y x   However, for stationary points of inflexion, 2 2 d 0d xa y x   Learning points 

SAJC 2020 JC1 H2 Mathematics Lecture Notes Chapter 4: Graphing Techniques Page 7 of 39 Therefore we can use the first derivative test. Stationary point of inflexion x a- a a d d y x + 0 + Sketch of tangent x a- a a d d y x – 0 – Sketch of tangent Example 3 (Worked) Find the exact stationary point(s) of 2e1xyx , if any, and state the nature of the stationary point(s). Solution 2d 2 e e e (2 )d x x xy x x x xx     At stationary points, d 0 e (2 ) 0d xy xxx      x = 0 or 2x  1y or 24e 1y  2 2 2 2 d 2e 2 e 2 e ed e (2 4 ) x x x x x y x x xx xx        Method 1: First Derivative Test x -1 0 1 d d y x 0 Sketch of tangent Therefore (0, – 1) is a minimum point. x -3 -2 -1 d d y x 0 Sketch of tangent Therefore   2