TMJC H2 Chapter 1 Graphing Techniques Discussion Questions_Solutions_2024

Chapter 1 Graphing Techniques TMJC 2024 Page 1 of 20 H2 Mathematics (9758) Chapter 1 Graphing Techniques Discussion Questions Solutions Level 1 1 The curve 1C and 2C have equations e1xy −=+ and ( )ln 2yx=+ respectively. (i) Sketch 1C and 2C on the same diagram, stating the exact coordinates of any points of intersection with the axes and the equations of any asymptotes. (ii) Use your calculator to determine the coordinates of the intersection point(s). 1 Solution (i) e1xy −=+ As x→ , 1y→ Therefore, 1y= is a horizontal asymptote. As x→− , y→ ( )ln 2yx=+ As 2x→− , y→− Therefore, 2x=− is a vertical asymptote. (ii) Using GC, the intersection point is ( )1.44,1.24 to 3 s.f y x O

Chapter 1 Graphing Techniques TMJC 2024 Page 2 of 20 2 Write down the equations of asymptotes of the following graphs: (a) 2 1y x= − (b) 72 3y x=− + (c) 11 2yx x= − − − Hence, sketch the graphs on separate diagrams, indicating equations of asymptotes and coordinates of axial intercepts and turning points (if any). 2 Solution (a) Asymptotes: 1, 0xy== (b) 72 3y x=− + Asymptotes: 3, 2xy=− = y x y = 0 x = 1 𝑦 = 2 1 − 𝑥 (0, 2) y x y = 2 x = −3 ൬0, − 1 3൰ ൬1 2 , 0൰ Recall 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)

Chapter 1 Graphing Techniques TMJC 2024 Page 3 of 20 (c) Asymptotes: 2, 1x y x= = − Note For questions that require exact coordinates, students need to find axial intercepts by algebraic method (without GC). i.e. Put 0y= to find x-intercept and put 0x= to find y-intercept. For this question, the x-intercepts are 35 ,02  − and 35 ,02  + . 3 Sketch, on separate diagrams, the following graphs: (a) 22( 3) ( 4) 25xy− + + = (b) 2 2 14 yx += (c) 22 14 16 yx −= (d) ( ) 2 22 14 x y− −= indicating clearly the main relevant features of the graph. Q3 Solution (a) y x x = 2 ( )0,0O ( )0, 8− ( )3, 4− 5 y x ( )6,0 22( 3) ( 4) 25xy− + + = For circle, label - Centre - Radius O ( )0.382,0 10, 2  − Final answer must be 3 s.f. for non-exact values.

Chapter 1 Graphing Techniques TMJC 2024 Page 4 of 20 (b) 2 2 2 2 22114 1 2 y x yx + =  + = (c) 22 22 22 14 16 124 yx yx −=  − = Oblique asymptotes: 22 22 22 22 24 24 1 2 0yx yx yx −= = = 1 2yx= and 1 2yx=− For ellipse, label - Centre - Length of both horizontal/vertical semi-major/semi-minor axis For hyperbola, label - Centre of hyperbola (Intersection of oblique asymptotes) - Coordinates of vertices - Equation of oblique asymptotes To find equation of oblique asympt