SAJC Chapter 4 Graphing Techniques Teacher Copy

SAJC 2022 JC 2 H1 Mathematics Graphing Techniques Page 1 of 52 Chapter 4 (Pure Mathematics): Graphing Techniques Objectives At the end of the chapter, you should be able to: Content 4.1 Functions 4.2 Basic Graphs 4.2 Characteristics of a Graph 4.2.1 Axes Intercepts 4.2.2 Asymptotes 4.2.3 Stationary Points 4.2.4 Restrictions on values of x and y 4.2.5 Symmetry 4.3 Conics Section 4.3.1 Circles 4.3.2 Ellipse 4.3.3 Using GC 4.3.4 General Forms of Conic Relevant websites 1. Applet on circles and ellipses: http://mathinsite.bmth.ac.uk/applet/ellipses/ellipses.html 2. Applet on graphs of various functions: http://www.analyzemath.com/ Introduction For concept learning systems, graphical data representation is crucial. A good representation might make possible the learning of a concept that was not learnable using other representations. For example, in the case of the earthquake domain (earthquake data such as its location, epicenter, intensity, depth, type, etc.), w e need important information such as the distance between the earthquakes’ epicentres and the difference in time between the earthquakes. This information could make possible the learnin g of an important concept. This is known as a graph-based concept learning system. Below are some examples of “graphs” you see in physical world. (a) understand the characteristics of graphs with the help of a graphic calculator, locate the turning points, and determine the asymptotes (horizontal and vertical), axes of symmetry, and restrictions on the possible values of x and y; (b) select the appropriate “window” of a graphic calculator that would display the critical features of the functions when sketching graphs; (c) use a graphic calculator to draw and compare the graphs of a variety of functions; (d) understand the relationship between a graph and its associated algebraic equation.

SAJC 2022 JC 2 H1 Mathematics Graphing Techniques Page 2 of 52 4.1 Definition of Function A function is a rule or relationship where for every input there is only one output. An example of a function is 2 f ( ) 5xx=+ , where x is the input to the function f ( )x and 2 5x + is the output of the function f ( )x . For every input value of x, there is only one output value. In short, a function is a process. Example 1 For each of the mappings below, state, with a reason, whether it is a function. (i) Yes, since every input to function f has only one output. i.e. ( )f2 xx=+ (ii) Yes, since every input to function f has only one output. (iii) No, since 2 has no output. (iv) No, since 3 has two outputs. 0• 1• 2• 3• • 2 • 3 • 4 • 5 X Y 0• 1• 2• 3• • 1 • 2 • 4 X Y 2• 1• 2• 3• • • 2 • 4 X