ACJC 2025 Graphs and Transformations Lecture Notes

1 1 Graphs and Transformations SYLLABUS  Use of a graphing calculator or a graphing software to graph a given function  Important characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes of the following: 2y ax ; 2x by ; 2 2 2 2 1x y a b  ; 2 2 2 2 1x y a b  ; 2 2 2 2 1y x b a  ; ax by cx d   ; 2ax bx cy dx e     Equations of asymptotes, axes of symmetry, and restrictions on the possible values of x and/or y  Effect of transformations on the graph of f ( )y x as represented by f ( )y a x , f ( )y x a  , f ( )y x a  and f ( )y ax , and combinations of these transformations  Relating the graphs of f ( )y x ,  fy x and 1 f ( )y x to the graph of f ( )y x  Simple parametric equations and their graphs

ACJC 2025/2026 H2 Mathematics (9758) 2 CONTENTS 1 Graphs of Conic Sections ................................ .............................. 3 1.1 Graphs of Parabolas ................................ ............................. 4 1.2 Graphs of Ellipses ................................ ................................ 5 1.3 Graphs of Hyperbolas ................................ .......................... 8 2 Characteristics of Graphs ................................ ............................ 11 2.1 Intersections with Coordinate Axes ................................ .... 11 2.2 Asymptotes ................................ ................................ ........ 11 2.3 Stationary Points ................................ ................................ 14 2.4 Symmetries ................................ ................................ ........ 14 3 Systematic Curve Sketching ................................ ........................ 15 4 Graphs of Rational Functions ................................ ...................... 18 4.1 Graphs of ax by cx d   , 0c ................................ ............... 18 4.2 Graphs of 2ax bx cy dx e    , 0a , 0d  ........................... 20 5 Simple Transformations ................................ .............................. 21 6 The Sequence of Transformations ................................ ............... 26 7 Graphs of Modulus Functions ................................ ..................... 30 8 Graph of 1 f ( )y x ................................ ................................ ...... 32 9 Parametric Equations and Their Graphs ................................ ....... 40 Annex A: Basic Standard Graphs ................................ ......................... 46 Annex B: Graphing Calculator Activities ................................ ............. 50 Annex C: Practice Questions ................................ ................................ 60

1 Graphs and Transformation 3 LECTURE 1 Lesson Outline  Review of standard graphs & characteristics (Annex A)  Graphs of conic sections: parabolas & ellipses 1 GRAPHS OF CONIC SECTIONS The study of the set of curves known as conics is a good example of how mathematicians have the knack for studying something so seemingly useless, yet turned out to be of immense importance to the world centuries later. Menaechmus (c. 375 -325 BC), tutor to Alexander the Great, conceived the idea of conic sections in attempts to solve construction problems such as that of the famous geometrical problem of “doubling the cube”. He was unsuccessful, and conic sections were cast aside until much later. In the early 17th century, Johannes Kepler, a German mathematician, astronomer, and astrologer, best known for his laws of planetary motion, brought conics to life again. He discovered that the planets travel around the sun in an elliptical orbit (Kepler’s First Law!). Galileo’s usage of parabolas in his work on projectile motion allowed cannoneers in the 17th century to work out approximately where their cannon balls will land when fired at a specific angle. Today, we utilise conic sections in a variety of applications, often employing their reflective properties in mirrors and satellite dishes. We shall now study the basic properties of conic sections. The general equation of any conic section is of the form 2 2 0ax bxy cy dx ey f      , where a, b, c, d, e and f are constants and a, b and c are not all zero. We can identify different types of conic sections by observing the coefficients of the variables in the equation and/or rearranging the equation. Some of the conic sections you will learn in this chapter are parabolas, ellipses, circles and hyperbolas.

ACJC 2025/2026 H2 Mathematics (9758) 4 1.1 Graphs of Parabolas Graphs of the form 2y ax ( 0a ) are parabolas with vertex at the origin and are symmetrical about the x-axis. Graphs of the form 2x ay ( 0a ) are parabolas with vertex at the origin and are symmetrical about the y-axis. Example Sketch the graphs with the equations 2y ax and 2x ay , where 0a . Solution ■ Graphs of the form     2 y k a x h   , where 0a , are parabolas with vertex (h, k) and are symmetrical about the line y k . Graphs of the form     2 x h a y k   , where 0a , are parabolas with vertex (h, k) and are symmetrical about the line x h . y x O y x O y x O y x O y x O y x O

1 Graphs and Transformation 5 Self-Practice Exercise 1 Sketch the graphs given by the equations     2 y k a x h   and     2 x h a y k   , where 0a . Solution ■ 1.2 Graphs of Ellipses Graphs of the form 2 2 2 2 1x y a b  , where 0a , 0b , are ellipses with the origin as the centre. These ellipses are symmetrical about the x- and y-axes. Graphs of the form 2 2 2 2 ( ) ( ) 1x h y k a b     are ellipses with centre ( h, k). These ellipses are symmetrical about the lines x h and y k . The above result can be obtained using simple transformations (see Section 5). y x O y x O b y x O y x O a b a b y x O y x O a b a

ACJC 2025/2026 H2 Mathematics (9758) 6 Example 1 Sketch the graphs of (a) 2 2( 1) 19 4 x y   , (b) 2 24 6 16 0x y y    , (c) 2( 2) 4( 1)y x   . Solution (a) 2 2( 1) 19 4 x y   (b) 2 24 6 16 0x y y    (c) 2( 2) 4( 1)y x   ■ Special case: a = b (Graphs of circles) Circles are special cases of ellipses, just as squares are special cases of rectangles. When a b r  , the general equation of the ellipse becomes     2 2 2x h y k r    , which is the equation of a circle with centre at (h, k) and radius r. Alternatively, it can be given in the general form 2 2 0ax ay cx dy e     where 0a . Compare this with the general equation of an ellipse 2 2 0ax by cx dy e     where 0a and 0b . y x O y x O y x O When the equation is in “expanded” form, we complete squares to obtain the standard form. This equation only has a y2 term and no x2 term. Therefore it is a parabola.

1 Graphs and Transformation 7 Example 2 Sketch the graph of 2 2 6 2 1 0x y x y     . Solution 2 2 6 2 1 0x y x y     ■ y x O When the equation is in “expanded” form, we will complete the square to obtain the “standard” form to determine the features of the conic.

ACJC 2025/2026 H2 Mathematics (9758) 8 LECTURE 2 Lesson Outline  Graphs of conic sections: hyperbolas  Characteristics of graphs: axial intercepts and asymptotes 1.3 Graphs of Hyperbolas Graphs of the form 2 2 2 2 1x y a b  or 2 2 2 2 1y x b a  are hyperbolas. These hyperbolas are symmetrical about both the x- and y-axes. Linear Asymptotes When a graph approaches a particular straight line as the value of x or y tends to infinity, then this straight line is called