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 Y 0• 1• 3• • 1 • 2 • 4 • 9 X Y

SAJC 2022 JC 2 H1 Mathematics Graphing Techniques Page 3 of 52 4.2 Basic Graphs (A) Polynomial Graphs Graphs with equation in the form 1 1 1 0 nn nny a x a x a x a − −= + + + + where 0na  and n is a positive integer. (i) y ax b=+ (Linear Functions) Linear functions have equations in the form y ax b=+ where a is the gradient and b is the y-intercept of the line. Example: 23yx=+ ( 0a ) Example: 24 33 xy=− + ( 0a ) (ii) 2y ax bx c= + + where 0a (Quadratic Functions) Quadratic functions have equations in the form 2y ax bx c= + + , where 0a . The graph of a quadratic function is a parabola symmetrical about a vertical axis. It is easier to sketch a quadratic function in completed square form, i.e. ( ) 2 y a x p q= − + . Note: (i) For 0a  , the curve has a minimum point (“U - shape”). For 0a  , the curve has a maximum point (“Inverted U - shape”). (ii) The maximum/minimum point has coordinates ( ),pq . (iii) The x-intercepts (if any) and y-intercept can by found by fixing 0=y and 0=x respectively. y x x 0 23yx=+ y 0 24 33 xy=− + 3 ,02 − ( )0,3 03xy=  = 30 2yx=  =− 40, 3   ( )2,0 240 33 2 xy x =  = =

SAJC 2022 JC 2 H1 Mathematics Graphing Techniques Page 4 of 52 (iv) The x-intercepts (if any) are given by a2 ac4bb 2−−=x . Example: 231yx=− ( 0a ) Example: 28 5 3y x x= + − ( 0a ) 22 2 2 2 58 5 3 3 8 3 5 5 5 13 8 3 106 6 6 12 y x x x x xx = + − =− − +       =− − − + =− − +           (iii) 32y ax bx cx d= + + + (Cubic Functions) Cubic functions have equations in the form 32y ax bx cx d= + + + , x , where 0a . Example: 3 1yx=+ ( 0a ) Example: ( )( )11y x x x= − + ( 0a ) y y x x 3 1yx=+ x (1 )( 1)y x x x= − + 231yx=− 28 5 3y x x= + − 0 0 0 0 y y 1 ,0 3 − 1 ,0 3   ( )0,8 ( )1,0− 22 ,03   ( )1,0− ( )0,1 ( )1,0− ( )1,0 ( )min point : 0, 1− 5 11max point : ,56 12   ( )( ) ( ) 2 08 0 3 5 8 0 5 25 4 3 8 5 121 16 or 12 3 6 6 8 or 13 xy y x x x =  = =  − − = −  − − −= = = − =−

SAJC 2022 JC 2 H1 Mathematics Graphing Techniques Page 5 of 52 (B) Reciprocal Functions n ay x= where n is a positive integer Reciprocal functions have equations in the form 1y x= , x , 0x . The graph tends towards the asymptotes 0=x and 0=y . (i) 2y x= 0a 0a (ii) 2 ay x= 0a 0a Note: 1. The graphic calculator has to be in function mode to graph f ( )yx= . 2. GC will not show you the asymptotes, axes intercepts and max and min points. Therefore, DO NOT copy the graph from the GC blindly. y x y x 2 3 xy= xy 2= y y x x 0 0 0 0 0y= 0y= 0y= 0y= 0x= 0x= 0x= 0x=

SAJC 2022 JC 2 H1 Mathematics Graphing Techniques Page 6 of 52 4.2 Characteristics of a Graph The following are important features of a graph to take note: (A) Axial Intercepts (B) Asymptotes (C) Stationary Points (D) Symmetries 4.2.1 Axial Intercepts An x-intercept of a graph is a point where the graph intersects the x-axis. To find the x-intercepts of a graph, let 0y= and solve the equation for x. A y-intercept of a graph is a point where the graph intersects the y-axis. To find the y-intercepts of a graph, let 0x= and solve the equation for y. Example 2 Find the axes intercepts of the graph 1 .2 xy x += + Solution: 1When 0, . 2xy== When 0, 1yx= =− . Therefore the x-intercept is ( )1,0− and y-intercept is 10, 2   . x y x-intercept y-intercept

SAJC 2022 JC 2 H1 Mathematics Graphing Techniques Page 7 of 52 Example 3 (Using GC) Find the axial intercepts of 2e1xyx=− . Solution: When x = 0, 1y=− . When y = 0, 2e 1 0xx −= (Using GC) This equation is difficult to solve by using algebraic method. We use GC to determine x-intercept. Using GC to find x-intercept Steps Screenshot Notes 1. Press y/ Select 2: zero or simply press Á 2. Respond to “Left bound?”, move the cursor to the left of the x-intercept and press Í 3. Respond to “Right bound?”, move the cursor to the right of the x-intercept and press Í 4. When prompted “Guess?”, press Í 5. The x-intercept is 0.7034674x= .

SAJC 2022 JC 2 H1 Mathematics Graphing Techniques Page 8 of 52 Using GC to find y-intercept Steps Screenshot Notes 1. Press y/ Select 1: value Press Í 2. Enter the value x = 0. Press Í 3. The y-intercept is (0, – 1 ) 4.2.2 Asymptotes (a) Vertical Asymptotes 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.