TJC 2025 IP4 Math Unit 16 Graphing with GC Lesson 2 (Student)

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 1 Temasek Junior College (IP) 2025 Year 4 Mathematics Unit 16 – Graphing with Graphing Calculator (GC)1 Lesson 2: Analysing Graphical Behaviour Learning Objectives By the end of this lesson you should be able to: 1) Sketch the graphs of different functions with the aid of a graphing calculator (GC). 2) Analyse behaviour of the graphs of different functions with the aid of a GC. 3) Solve simple problems involving graphs using the GC. 2.1 Asymptotic Behaviour Consider graphing the function 1 11y x=+ − using the GC. Recall: • The Y= button will bring you to the graphing interface. • To enter fractions use ALPHA Y= 1. • To enter the variable (x), use X,T,θ,n. • To generate the graph of a function, use GRAPH after entering the function. The screen capture above is how the graph of 1 11y x=+ − will look like in the ZStandard ( ZOOM 6 ) setting. Notice that the asymptotes are not displayed. 1 This set of notes is to be used in conjunction with the TI-84 Plus CE graphing calculator. Operation of other brands and models of graphing calculators may differ AM

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 2 In fact, the GC is not able to display asymptotes. Any asymptotes will have to be drawn based on your understanding of the graph and the function it represents. In other words, you SHOULD NOT BE RELIANT on the GC to s how every feature of the graph of a function. That being said, the GC can still be used to provide an idea of what the asymptotes might be. This needs to done in conjunction with: • Your knowledge of the graph and the function it represents. • The visual observations made on the graph displayed using appropriate WINDOW and ZOOM settings (we shall look at this at a later stage of the lesson). • An understanding of the definition of different types of asymptotes. Analysing horizontal asymptotic behaviour Recall that a horizontal asymptote is a line with equation yk= , where k is the value a function approaches as x approaches + and − , and may or may not be the same at + and − . With this understanding, we can use the GC to observe what happens to the value of a function when x is an extremely large positive value and an extremely negative value. This will allow us to identify possible asymptotic behaviour(s). To determine what happens to 1 11y x=+ − when x becomes an extremely large positive value: • Press 2ND Window to access the TABLE SETUP of the GC. • Enter an extremely large positive TBLSTART value of x for e.g. 1 2ND , 9 9, which is 1E99 i.e 9910 . • Press 2ND GRAPH and observe the values. Change this value

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 3 • Observations ✓ The values of x simulates the event where x approaches + . ✓ The values of y are observed to “constantly” be 1y= , which is the value y approaches when x approaches + . ✓ There is a horizontal asymptote of 1y= . In fact, our current knowledge of the function should tell us that the values are approaching the value 1 and will never reach the exact value 1. Try it! Use your GC to analyse the asymptotic behaviour of 1 11y x=+ − when x approaches − . Analysing vertical asymptotic behaviour Recall that a vertical asymptote is a line with equation xk= , where k represents the value(s) of x for which the function is undefined. With this understanding, we can use the GC to identify the vertical asymptotes. • From the graph, we know that there is possibly a vertical asymptote around 1x= . Each division on the x-axis is 1 (under ZStandard, ZOOM 6 settings). You can verify this using XSCL in the WINDOW. It may be different under different WINDOW and ZOOM settings. • Under TABLE SETUP ( 2ND WINDOW ), enter a TBLSTART value close to 1 (or where the asymptote is observed to be located on the graph) e.g. 0. • We then access the tabulated values ( 2ND GRAPH ) to identify value(s) of x for which the function is undefined. Determines what a division represents.

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 4 • Observations ✓ The value of y is undefined when 1x= (indicated as ERROR). ✓ There is a vertical asymptote of 1x= . Important! • The possible location(s) of the vertical asymptote(s) may not be displayed on the screen when the WINDOW and ZOOM settings are inappropriate. For example, the possible location of the v ertical asymptote of the graph of 1 15y x= − , will not be displayed under ZStandard ( ZOOM 6 ) settings. • A graph may have more than 1 vertical asymptote and not all may be displayed simultaneously e.g. ( )( ) 1 1 15y xx= −− under ZStandard ( ZOOM 6 ) settings. Adjusted WINDOW settings ZStandard settings Adjusted WINDOW settings ZStandard settings

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 5 • A suitable incremental interval of x ( TBL ) is required in the TABLE SETUP to subsequently identify the value(s) of x for which a function is undefined. For example, using an incremental interval of 1 for x ( TBL = 1), we will not be able to identify the value of x for which 1 1.5y x= − is undefined. Consequently, the vertical asymptote cannot obtained. In this case, an incremental interval of 0.1 is more appropriate. • While the GC can help us locate the vertical asymptote s, a lack of knowledge of the function will still not allow us to identify all of them, especially when they are not displayed. In this case, one will not be able to find the un-displayed asymptotes without good knowledge of the function. • In other words, you SHOULD NOT BE RELIANT on the GC to show every feature of the graph of a function. Mastering the mathematical knowledge required is necessary. ERROR detected Appropriate interval Unable to detect ERROR Inappropriate interval

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 6 Exercise 1 Sketch the graph of 1 11y x=+ − . Discussion (a) Write down the equation of the asymptote of the graph of lnyx= . (b) Without the use of a GC, sketch the graph of lnyx= . (c) (i) Now, graph lnyx= on your GC. (ii) What do you observe has occurred? Why do you think this has occurred? (iii) What lessons can you learn? (a) 0x= (b) (c)(i) (c)(ii) The graph of lnyx= seemed to have “truncated” in the GC display. This is due to the display limitations of the GC, which can be improved with different ZOOM settings. (c)(iii) Again, we cannot be totally reliant on the GC to tell us all the features of the graph. In this case, a lack of knowledge of the graphs of logarithmic functions will result in us being misled by the display seen on the GC. O 1

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 7 2.2 The ZOOM and WINDOW Settings Example 1 With the aid of a GC, (i) sketch the functions ( )ln 2yx= and 2 8 18y x x= − + on the same set of axes. (ii) find all the solutions to the equation ( ) 2ln 2 8 18x x x= − + . [Solution] (i) This will probably be what you see on the GC using ZStandard ( ZOOM 6 ) settings. To get a better view of the intersections, we can use the ZBOX function ( ZOOM 1 ). • Select ZOOM 1 to activate the ZBox function. • Move the cursor to a position near the left (and slightly above) of the intersection point using the arrow keys. • Press ENTER, t hen scroll rightwards and downwards to form a box around the intersection point. • Press ENTER again to zoom into the boxed up portion of the graph. It is unclear how many intersections exists between the two graphs. Move the cursor near the intersection(s)

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Pag