ASR Standard Curves Lecture Notes

JC1 H2 Mathematics Chapter 01: Standard Curves Standard Curves C01 – 1 ANDERSON SERANGOON JUNIOR COLLEGE 2026 JC1 H2 MATHEMATICS CHAPTER 01: STANDARD CURVES Lesson Objectives At the end of the chapter, you should be able to ▪ relate the following equations: y = |x|, 22 22 1xy ab+= , 22 22 1xy ab−= , 22 22 1yx ba−= and equations of parabolas with their graphs, identifying their characteristics such as symmetry, axial intercepts, turning points, asymptotes and restrictions on the possible values of x and/or y. ▪ use the graphic calculator to graph a curve with equations stated above. ▪ sketch simple parametric equations, and convert parametric equation into its Cartesian equation. Pre-Requisites You should already know: Standard graphs learnt previously at the O -Levels, such as straight line s, quadratic curve s, circles, the logarithmic curve (i.e. lnyx= ) and the exponential curve (i.e. exy= ). Pre-Lesson Activity • Watch video ‘Using the Conics App’ on https://education.ti.com/en/resources/test- preparation#lightbox=Test-Prep-tips-for-TI-84-Plus-CE to familarise with GC keystrokes. • Read Section 1 and complete exercises (1) to (3). (1) Find the equation of the line which passes through the point (2, ‒3) and makes an angle of π 3 with the positive x-axis. [Solution] [y = 3 2 3 3x −− ] (2) Sketch the graph of 1 24e x y=− , giving the exact coordinates where the curve cuts the axes, and the equation of the asymptotes. State any increasing and/or decreasing properties. [Solution] [(4ln 2, 0), (0, 3), y = 4] (3) Sketch the curve of 2 42y x x=− + − , indicating the coordinates of intercepts and the turning point. [GC steps in Annex 1]

JC1 H2 Mathematics Chapter 01: Standard Curves Standard Curves C01 – 2 Section 1: Standard Curves 1.1 Polynomial Curves Type of Graph Common Form of Equation Graph Straight Lines ▪ y = mx + c; or ▪ ( )00y y m x x− = − , where m is the gradient, c is the y-intercept and ( )00,xy is a point on the line. Note (1) 10 10 or tanyym xx −= −  , where  is the angle that the line makes with the positive x- axis in the anti-clockwise direction. (2) Vertical line: xh= Horizontal line: yk= Quadratic Curves ▪ y = ax2 + bx + c (a  0) a > 0 a < 0 Cubic Curves ▪ y = ax3 + bx2 + cx + d (a  0) a < 0 a > 0  y m > 0 • x c x • y  O m < 0 c Note: tanm=  Equivalence - is a relationship that express the equality of 2 math objects. In this case, the gradient of a straight line can be represented when you have either 1. coordinates of 2 points, or 2. Angle that the line makes with the positive x-axis. BIG IDEA

JC1 H2 Mathematics Chapter 01: Standard Curves Standard Curves C01 – 3 1.2 Exponential and Logarithm Curves Type of Graph Common Form of Equation Graph Properties Exponential Curves , where 0 and 1 xya aa =  If 1a , then xa increases at an increasing rate, without limit If 01 a , then xa decreases and approaches 0 Asymptote: y = 0 e (base e 2.72 0)kxy=   If 0k  , then ekxy= is increasing at an increasing rate If 0k , then ekxy= is decreasing and approaches 0 Asymptote: y = 0 Logarithm Curves ln , 0y x x= Asymptote: x = 0 Relationship between lnyx= and exy= : Reflection in the line y = x x y 0 1 y = ax a > 1 0 < a < 1 y = 0 x y 0 1 y = ekx k > 0 k < 0 y = 0 x y 0 1 y = ln x x = 0 Diagrams – Cartesian graphs provide a way to visualize the behavior of relationship between two variables. It highlights features of the graphs such as showing the asymptotic behavior of a function/ cartesian equation BIG IDEA

JC1 H2 Mathematics Chapter 01: Standard Curves Standard Curves C01 – 4 1.3 Trigonometric Curves Type of Graph Equation Graph Properties Trigonometric Curves siny=  (1) – 1  sin   1 (2) Period of 0360 [or 2]. (3) Symmetrical about the origin. cosy=  (1) – 1  cos   1 (2) Period of 0360 [or 2]. (3) Symmetrical about the y-axis. tany=  (1) − < tan  < . (2) Vertical asymptote at  = …, 3 2−  , 2− , 2  , 3 2 , … (3) Period of 180 [or ] 1sinyx −= Principal range is 111 sin22 x−−   1cosyx −= Principal range is 10 cos x−  1tanyx −= Principal range is 111 tan22 x−−   –2 – 0  2  y –1 1 –2 – 0  2  y –1 1 –2 – 0  2  y –1 1 /2  y –1 1 –/2   y –1 0 1 /2  y 0 –/2

JC1 H2 Mathematics Chapter 01: Standard Curves Standard Curves C01 – 5 1.4 The Modulus Function y = |x| Definition: The modulus of a real number x, denoted by x , is the non-negative numerical value (or absolute value) of x, i.e. the value of x without regard to its sign. Significance of |x| x = distance of the point representing x from the point representing 0 on the number line. Basic Properties: Property Example 1 if 0 if 0 xxx xx =− • |3| = 3 • |−2| = 2 2 0 x  for all real values of x 3 aa−= • 2 2 2− = = 4 ..a b a b= • 2 2 2x x x=  = 5 aa bb= • 222 aaa == 6 222 x x x== • 222 2 4== 7 2 xx = • 2 33 = but ( ) 2 3 3 3− = = − 8 If a is a positive constant, then x a= = xa • If | x | = 3, then x =  3 0 x Distance =|x| Distance =|x| -x y y = |x| x 0

JC1 H2 Mathematics Chapter 01: Standard Curves Standard Curves C01 – 6 Example 1 Sketch (2 1)( 2)y x x= + − , clearly labeling the axial intercept and turning point. [Solution] GC steps for sketching Modulus function (A) Press o. To enter the modulus function, press t and p for [f2], followed by 1: abs (. Key in the rest of the equation. (B) Key in the rest of the equation. Press s to view the graph. (Tip: If you cannot see the full graph, select zoom, to zoom in or out until you can see the whole graph) Refer to Annex A for keystrokes to find x intercept, y intercept and turning point. Test yourself Determine whether each of the following statements is ‘True’ or ‘False’. Explain why it is so and/or give the correct statement. 1 x y x y+ = + True/False 2 ( ) ( ) 2 2 2 1 2 1x x x x +   + True/False 1.5 Power Function Note: you need to label the turning point, and x and y intercept for a quadratic function.

JC1 H2 Mathematics Chapter 01: Standard Curves Standard Curves C01 – 7 A power function is a function ny ax= , where n is any real constant number. When n = 0, 1, 2 or 3, you will get a constant, linear graph, a quadratic graph and a cubic graph respectively. (Refer to Section 1.1) Type of Graph Common Form of Equation Graph Properties Graphs of ny ax= When n = – 2 and 1a= 2 1y x= Asymptotes 0y= , x = 0 When n = – 1 and 1a= 1y x= Asymptotes 0y= , x = 0 Note: An asymptote is a line which a curve approaches as x and y gets very large (i.e. x or y → ). You will learn more about this feature of graph in the chapter of Curve Sketching. 0 x y x = 0 0 x y x = 0

JC1 H2 Mathematics