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)  Symmetr