TMJC H2 Chapter 1 Graphing Techniques Learning Package 2024

Tampines Meridian Junior College 2024 H2 Mathematics (9758) Chapter 1 Graphing Techniques Learning Package Resources  Core Concept Notes  Discussion Questions  Extra Practice Questions SLS Resources  Recordings on Core Concepts  Quick Concept Checks

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Chapter 1 Graphing Techniques TMJC 2024 Page 1 H2 Mathematics (9758) Chapter 1 Graphing Techniques Core Concept Notes Success Criteria: Surface Learning Deep Learning Transfer Learning  Determine the equations of asymptotes: horizontal, vertical, oblique (if any) of exponential, logarithmic and rational functions.  Find, where applicable, the important features (ASMILE) of a graph.  Identify restrictions on possible values of x and/or y (if any)  Sketch the graph of a rational function such as: ax by cx d += + , 2ax bx cy dx e ++= +  Identify shapes of standard conic curves and state important characteristics of them: ( ) ( ) 2 y k a x h− = − or ( ) ( ) 2 y k a x h− = − ( ) ( ) 22 2x h y k r− + − = ( ) ( ) 22 22 1x h y k ab −− += , ( ) ( ) 22 22 1x h y k ab −− −= or ( ) ( ) 22 22 1y k x h ba −− −= .  Explore the conic APP in the GC to graph a conic section and note the limitations of the conic APPS  Use a GC to graph a given piecewise function  Use a GC to graph a given modulus function  Use a GC to graph a pair of parametric equations  Use GC to find intersection points of graphs  Draw the graph of a given function and label important characteristics such as asymptotes, turning points and intersections with the axes.  Show algebraically that a function cannot lie between 2 certain values (to be determined) and use this information to draw the graph  Convert conic section into standard form (using completing the square or long division)  Use GC (not conic APP) to graph a conic section  Draw the graph of a given piecewise function  Express modulus function as a piecewise function  Draw the graph of a function given in parametric form  Convert the equation of a curve between parametric and Cartesian forms  Sketch graphs of functions with unknowns  Manipulate expressions and make inferences about intersections between graphs Definition of Learning (John Hattie): The process of developing sufficient surface knowledge to then move to deeper understanding such that one can appropriately transfer this learning to new tasks and situations.

Chapter 1 Graphing Techniques TMJC 2024 Page 2 §1 Features of Graphs When sketching graphs, it is important to pay attention to important features. Go through the checklist and take note of each of the following whenever you sketch a graph. 1. Asymptotes (if any) 2. Shape of the graph 3. Maximum/minimum turning points (if any) 4. Intercepts with the axes (if any) 5. Labelling of the graph and axes 6. End points (if any) 1.1 Asymptotes Consider the following curves: (a) e2xy −=+ Observe that the curve seems to tend towards some value of y for large values of x. Consider: As x→ , e0x− → , 2y→ In this case, the line 2y= should be included in the graph. This line is as known as a horizontal asymptote of the curve. (b) ( )ln 3yx=+ Observe that the curve seems to tend towards some value of x for “large” negative values of y. Consider: As 3x→− , ( )ln 3x+ →− , y→− In this case, the line 3x=− should be included in the graph. This line is as known as a vertical asymptote of the curve. y O x y O x ( )2,0−

Chapter 1 Graphing Techniques TMJC 2024 Page 3 Given that the equation of a curve ( )fyx= : Vertical asymptote When xa→ , y→ then xa= is a vertical asymptote. For e.g. 0x= is a vertical asymptote to the graph of lnyx= . Horizontal asymptote When x→ , yb→ then yb= is a horizontal asymptote. For e.g. 0y= is a horizontal asymptote to the graph of exy= . Oblique asymptote When x→ , (note: 0)y ax b a→ +  then y ax b=+ is an oblique asymptote. Example 1 (Determine equations of asymptotes) State, with a reason, the equations of the asymptotes of the following graphs: (a) 41 2y x=+ − . Determining vertical asymptote(s): As 2x→ , 4 2x →− , y→ Vertical asymptote: 2x= Determining horizontal/oblique asymptote(s): As x→ , 4 02x →− , 1y→ Horizontal asymptote: 1y= (b) 93 2yx x= − + + . Determining vertical asymptote(s): As 2x→− , 9 2x →+ , y→ Vertical asymptote: 2x=− Determining horizontal/oblique asymptote(s): As x→ , 9 02x →+ , 3yx→− Oblique asymptote: 3yx=− y O x x y O (−5, −11) (1, 1)

Chapter 1 Graphing Techniques TMJC 2024 Page 4 1.2 Shape, Maximum and Minimum Turning Points and Axial Intercepts Definitions y-intercept Set 0=x in equation and solve for the value of y . If by = , then (0, )b is the point where the graph crosses the y-axis. (0, )b is also called the y-intercept. x-intercept Set 0y= in equation and solve for the value of x . If xa= , then ( ),0a is the point where the graph crosses the x-axis. ( ),0a is also called the x-intercept. Maximum turning point Stationary point (point on curve where gradient is zero) at which the y-value is greater than any other y-value in the neighbourhood on either side of the point. Minimum turning point Stationary point (point on curve where gradient is zero) at which the y-value is lower than any other y-value in the neighbourhood on either side of the point. GC Skills : TMJC Infinitum GC Video Tutorials TMJC Mathematics Department has developed a series of GC video tutorials for the various H2 Mathematics topics. How to access? 1. Log into your ICON email account, 2. Go to https://sites.google.com/tmjc.edu.sg/infinitum-gc/home or scan the QR code on the right, 3. Click on o Graphing Techniques ~ to learn about zoom features, window settings, finding key features of a graph Do Question 1. Go through the video tutorials for Graphing Techniques or the GC keystrokes. Ensure that you are able to find the various key features (shape, maximum, minimum turning points, axial intercepts) of the curve.

Chapter 1 Graphing Techniques TMJC 2024 Page 5 Question 1 (Use of GC to obtain main features of a graph) The curve C is defined by 32 25y x x x= + − + . Find the coordinates of the (a) axial intercepts of C (b) maximum and minimum turning points of C. Sketch curve C. Answer: GC keystrokes to sketch curve GC Steps Screen on GC • Press o and key in the equation of the curve. • Press the following keys to input the given equation: „›Â~ÃÁ„¡¹ „÷ • To obtain the sketch of the graph, press s. Note: To reset to standard window setting before sketching the graph, press q and select 6: ZStandard. O Maximum turning point Minimum turning point y x y - intercept x - intercept Note: Round off to 3 significant figures for non-exact answer.

Chapter 1 Graphing Techniques TMJC 2024 Page 6 GC keystrokes to obtain turning points GC Steps Screen on GC • To obtain the turning points, use the CALCULATE function by pressing yr. • To obtain the maximum point, Press ¶ to select 4:maximum. • GC will prompt for a left bound. Press ~ or | to move the cursor slightly to the left of the maximum point and press Í. • GC will prompt for a right bound. Press ~ or | to move the cursor slightly to the right of the maximum point and press Í. • GC will prompt for a Guess. • Press Í. GC will return the maximum point of ( )1.55,7.63− (3 s.f.) • To obtain the minimum point, Press  to select 3:minimum. Repeat the steps stated above to obtain the minimum point. Maximum turning point: ( )1.55,7.63− (to 3 s.f.) Minimum turning point: ( )0.215, 4.89 (to 3 s.f.)

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