RI C8A Graphing Techniques Lect Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ___________________________ Chapter 8A: Graphing Techniques Page 1 of 40 Chapter 8A: Graphing Techniques SYLLABUS INCLUDES  Equations of asymptotes, axes of symmetry, and restrictions on the possible values of x and/or y  Graphical Interpretation of (i) f( ) 0x  , f( ) 0x  and f( ) 0x  (ii) f() 0x  and f() 0x   Determining the nature of the stationary po ints (local maximum and minimum points and points of inflexion) analytically, in simple cases , using the first derivative test or the second derivative test  Locating maximum and minimum points using a graphing calculator  Important characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes of the following: 2ya x ; 2xb y 22 22 1xyab 22 22 1xyab ; 2222 1yxba ax by cx d  2ax bx cy dx e   Use of a graphing calculator or a graphing software to graph a given function  Simple parametric equations and their graphs

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ ____________________________ Chapter 8A: Graphing Techniques Page 2 of 40 CONTENT 1 Characteristics of Graphs 1.1 Intersections with the axes 1.2 Axes of symmetry 1.2.1 Symmetry about the x-axis or y-axis 1.2.2 Symmetry about other lines 1.3 Restrictions on the possible values of x and/or y 1.4 Asymptotes 1.5 Shape of graphs 1.5.1 Monotonicity & Concavity 1.5.2 Stationary Points, Turning Points & Points of Inflexion 2 Rational Functions 2.1 Graphs of functions of the form f( ) ax bx cx d  , where 0c , and the numerator and denominator have no common factor if 0a 2.2 Graphs of functions of the form 2 f( ) ax bx cx dx e  , where 0, 0ad and the numerator and the denominator have no common factors 3 Conic Sections 3.1 Parabola 3.2 Circle 3.3 Ellipse 3.4 Hyperbola 4 Parametric Equations and their graphs Appendix I Sketching ellipse and hyperbol a using a graphing calculator

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ ____________________________ Chapter 8A: Graphing Techniques Page 3 of 40 y x 1 – 1 y x 2 – 2 O O 1 CHARACTERISTICS OF GRAPHS In sketching graphs, certain features of a curve can be observed, or easily calculated, from the equation of the curve. Some of the important features that we can consider are intersections with the axes, axes of symmetry, asymptotes, restrictions on the possible values of x and/or y turning points etc. 1.1 Intersections with the axes It is usually easy to determine whether the curve f( )yx meets the x-axis and/or y-axis. The x intercepts are found by setting 0y in the equation of the curve and solving for x. Similarly, the y intercepts are found by setting 0x in the equation of the curve and solving for y. For example, the curve with equation 22 149xy  clearly intersects the x-axis at (2,0) and (2 , 0 ) but does not cross the y-axis. 1.2 Axes of symmetry 1.2.1 Symmetry about the x-axis or y-axis If the equation of the curve is unchanged when x is replaced by  x , the graph is symmetrical about the y-axis. Similarly, if the equation of the curve is unchanged when y is replaced by  y , the graph is symmetrical about the x-axis. For example, the graph of the curve with equation 2ln( )yx is symmetrical about the y-axis while the graph of the curve with equation 22 149xy  is symmetrical both about the x-axis and y-axis. Note: The dotted lines 3 2yx and 32yx are asymptotes to the curve 22 149xy  . In the case of the curve 2ln( )yx , the line 0x is an asymptote to the curve. We will learn more about asymptotes in section 1.4.

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ ____________________________ Chapter 8A: Graphing Techniques Page 4 of 40 y x 1 1 y x 4 O 2 O 1.2.2 Symmetry about other lines Besides symmetry about the axes, the graph of the curve could possibly have other axes of symmetry. For the curve with equation  2 42yx  , the graph is symmetrical about the line 2x . In the case of the curve with equation 11 1y x  , the lines of symmetry are yx and 2yx  . Note: To determine if a curve is symmetrical in the line yx , we can easily do so by replacing both x with y and y with x. If the equation of the curve is unchanged, then the line yx is a line of symmetry. However, it may not be easy to see from its equation that the curve with equation 11 1y x  also has 2yx  as a line of symmetry. We will be able to understand this better in Section 5 (Standard Transformation). Exercise: State a line of symmetry of the following curves whose equations are (i) 2 8 1y xx  , (ii) 2 e xy  . Answer: (i) 12x , (ii) 0x

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ ____________________________ Chapter 8A: Graphing Techniques Page 5 of 40 1.3 Restrictions on the possible values of x and/or y It is sometimes useful to consider whether ther e are any restrictions on the possible values of x and/or y when we graph the curve f( )yx . For instance, the curve with equation  ln 2y x is only well-defined when 2x . Thus the graph of  ln 2y x lies only in the region where 2x . In the case of the curve with equation 2 1yx , we observe that since 2 0y  for all y  , then 10 1x x . Thus the graph lies only in the region where 1x . Example 1 It is given that that 2 3 ,, 11xyx x x     . Without using a calculator, find the set of values that y can take. Solution If y can take the value k, the graphs of 2 3 1 xy x   and yk must intersect. At the point(s) of intersection,  2 2 2 3 1 3 30 xk x kx k x xk x k      For the quadratic equation to have real roots,   2 Discriminant 0 41 2 0 62 0 kk kk     6o r 2kk  Hence, set of values y can take is   2.,6 ,    Note: Graphs of parabolas, circles, ellipses and hyperbolas (which will be dealt with in Section 3) have certain restrictions on values of x and y as well. Consider the graph of 22 9xy , which is a circle with centre at the origin and radius 3 units. We note that 33 x  and 3 3 y  . For the graph of hyperbola 2294 3 6xy , we observe that as 2293 6 4x y . Since 2 0y  for all y  , we have 2 4x  . Thus 2or 2xx  .

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ ____________________________ Chapter 8A: Graphing Techniques Page 6 of 40 1.4 Asymptotes An asymptote of the curve with equation f ( )yx is a line or a curve that the graph of f ( )yx approaches when x or y is large. It gives an indication of the behaviour of the curve when or x  and/or when or y  . In our syllabus, we will only concern ourselves with linear asymptotes. Example 2 Find the equations of the asymptotes of the curve with equation 12y x . Solution As ,2xy  and hence 2y is a horizontal asymptote. Note that, in fact, we also have as ,2xy  . As 0x  (i.e. approaching 0x from the left), y . Thus 0x is a vertical asymptote. Note also that as 0x 