1. 2022 Graphing Techniques (For Upload)

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: GRAPHING TECHNIQUES Page | 1 H2 MATHEMATICS TOPIC GRAPHING TECHNIQUES 2022/JC1 Content Outline  Use of a graphing calculator to graph a given function.  Sketching graphs with important characteristics such as symmetry, intersections with axes, turning points and asymptotes of the following:  ax bycx d  2ax bx cydx e   Determine the equations of asymptotes, axes of symmetry, and restrictions on the possible values of x and/or y. 1 Introduction Have you seen graphs (bar, pie, histogram, line) in newspapers, television, magazines and books etc.? The purpose of the graphs is to show numerical facts in visual form so that they can be understood quickly, easily and clearly. Thus graphs are visual representations of data collected. Data can also be presented in the form of a table; however a graphical presentation is easier to understand. In the H2 Mathematics syllabus, graphing calculator (G.C.) will aid in the sketching of graphs. However, students should be aware that there are limitations inherent in G.C. G.C. Basics: Before sketching a graph, ensure that the settings are as shown. Settings can be accessed by pressing mode . . (Use the arrow keys to navigate and enter key to highlight FUNCTION.)  Self-Reading G.C. can be used to sketch the graphs of the following basic curves: Equation Press Input Screen Result 2y x y= X,T,,n x2 graph brings us to the input screen for specifying equations displays the plotted graph (of the specified equation)

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: GRAPHING TECHNIQUES Page | 2   Equation Press Input Screen Result 3y x y= X,T,,n ^ 3 graph To zoom in at the point  0, 0, press : zoom 2 enter yx y= 2nd x2 X,T,,n graph 1yx y= 1  X,T,,n graph Alternative: Use alpha y= 21yx y= 1  X,T,,n x2 graph Alternative: Use alpha y= 3y x y= math 4 X,T,,n graph exy y= 2nd ln X,T,,n graph exy y= 2nd ln () X,T,,n graph 3 e e

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: GRAPHING TECHNIQUES Page | 3 Equation Press Input Screen Result lny x y= ln X,T,,n graph  Remark: Due to the limitation of G.C., there might be a difference between what we see on the zoom 6: ZStandard screen and the actual graph that we draw. Take for example lny x: We should be familiar with the other zoom functions keys available in the G.C. so that we can use them to obtain an optimal view of the graph. Nevertheless, we should take note that G.C. is simply a tool that is used to determine the general shape of a curve. 2 Features of Graphs In this section, we will investigate the following features of graphs:  Axial intercepts  Asymptotes  Turning points  Lines of Symmetry  Possible range of values of x and y 2.1 Axial Intercepts An axial intercept is a point at which a curve intersects with either axis (x-axis /y-axis). A point of intersection between the curve and the x-axis is an x-intercept, while a point of intersection between the curve and the y-axis is a y-intercept. x y O

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: GRAPHING TECHNIQUES Page | 4 Example 1: Find the coordinates of the axial intercepts of the curve 22 15y x x  . Solution: Method : Algebraic/Analytical To find x-intercepts: Substitute 0y,   22 15 05 3 05 or 3x xx xx    To find y-intercept: Substitute 0x,  20 2 0 1515y    axial intercepts are  5,0,  3,0 and  0, 15. Method : Using G.C. To find the x-intercepts: Press Screen Display Remarks 1 Press y=. Key 22 15x x  into Y1. 2 Press 2nd trace to access the CALCULATE menu. Select 2: zero to find the x-intercepts. 3 Press arrow keys to move the cursor to the left of first root followed by enter. OR You can just enter a number that is to the left of the first root.

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: GRAPHING TECHNIQUES Page | 5 Press Screen Display Remarks 4 Press arrow keys to move the cursor to the right of first root followed by enter. OR You can just press a number that is to the right of the first root. The left and right boundary should only contain one root, else G.C. may find the other root instead. 5 For guess?, you can move the cursor close to the first root, then press enter. G.C. uses a numerical method to find the root. In some situations, the answer given may be 4.9999999 and you will need to round off accordingly after verifying your answer. You can verify your answer by substituting 5x into 22 15y x x   and check that 0y. 6 Repeat steps 2  5 to find the other root. Before proceeding to find the y-intercept, notice that the G.C. did not show the curve in its entirety as the minimum turning point is not shown. We will need to change the window settings. Press Screen Display Remarks 1 Press window to access the window settings menu. The default window settings are: Xmin10, Xmax10, Ymin10, Ymax10. 2 For this question, we can set Ymin to be 20. Changing the window setting is a trial and error process, i.e. there are no “correct” values to set.

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: GRAPHING TECHNIQUES Page | 6 Press Screen Display Remarks 3 Press graph. The minimum point can now be displayed. 4 Press 2nd trace to access the CALCULATE menu. Select 1:value to find the value of y-coordinate given the x-coordinate. 5 Press 0 then enter.  Remark:  If question states explicitly to find exact coordinates of the axial intercepts, then method  algebraic/analytical must be used.  For method : using G.C., do not press the arrow key to trace on the curve and simply read off the values as it is not accurate.  Self-Practice 1: Find the exact coordinates of the axial intercepts for the following curves: (a) 13yx (b) 2 12xyx (c) 21xyx (d) 22 32x xyx . Answers: (a) y-intercept :10, 3    ; no x-intercept (b) y-intercept:10,2    ; x-intercept:1, 02    (c) y-intercept / x-intercept : 0, 0 (d) y-intercept:30,2    ; x-intercept:   3,0 , 1,0

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: GRAPHING TECHNIQUES Page | 7 2.2 Asymptotes 2.2.1 Vertical Asymptotes If there is a constant k such that along a curve, as ,x k y  or y , then the vertical line x k is a vertical asymptote of the curve. In many curves, the value(s) of x which makes the function y undefined indicates the presence of a vertical asymptote at that value(s). In such curves, it will never intersect with its vertical asymptote(s). 2.2.2 Horizontal Asymptotes If there is a constant k such that along a curve, as x or x, y k, then the horizontal line y k is a horizontal asymptote of the curve. The curve might intersect with its horizontal asymptote(s) (Refer to Example 8c and 8d). Example 2: Given the curve 113yx , find the equations of the asymptotes and sketch the curve. Solution: When x ______ , y is _______________. Explanation: As 3x, 3 3 0  , i.e. a very small positive value.  10. As 3x, 3 3 0  , i.e. a very small negative value.  10. Thus as 3x, y. Hence 3x is a vertical asymptote. As x , 130x, and 1y, hence 1y is a horizontal asymptote. 3 110    y = 1 x = 3 x y O undefined 2,0 20,3   

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: GRAPHING TECHNIQUES Page | 8  Remark: Asymptotes should be determined and drawn using dotted lines before sketching the curve.  Self-Practice 2: Find the equation(s) of the vertical asymptote(s) for the following curves: (a) 13yx (b) 2 12xyx (c) 21xyx