[H2 MATH] Chapter 2 - Graphing Techniques

SHALYN TAY COPYRIGHTED © A LEVEL H2 MATHEMATICS GRAPHING TECHNIQUES

CHAPTER ANALYSIS MASTERY EXAM WEIGHTAGE SHALYN TAY COPYRIGHTED ©

STANDARD GRAPHS CONICS GRAPHING TECHNIQUES PART I SHALYN TAY COPYRIGHTED ©

Standard Graphs SHALYN TAY COPYRIGHTED © Characteristics of Graphs When sketching a graph, it is necessary to indicate the following: 1. X- and Y- Axes 2. Origin 3. Co-ordinates of Stationary Points 4. Co-ordinates of Intercepts 5. Equations of Asymptotes 6. Co-ordinates of End-Points 7. Equation of Graph You may use the following acronym to help you with curve sketching: S Shape / Stationary Point I Intercepts A Asymptotes Intercepts To find y-intercept: sub x = 0 and solve for y To find x-intercept: sub y = 0 and solve for x Stationary Points Stationary Point Turning Point Point of Inflection Maximum Minimum Shape of Curve 𝑑𝑦 𝑑𝑥 𝑑2𝑦 𝑑𝑥2 ≤ 0 ≥ 0 0 0 0 −+ 0 +− 0 ++ 0 −− Asymptotes For proper function 𝑓 𝑥 = 𝐴(𝑥) 𝐵(𝑥) , vertical asymptote can be found by letting 𝑄 𝑥 = 0 For improper function 𝑓 𝑥 = 𝐴(𝑥) 𝐵(𝑥) , carry out long division to express it as 𝑓 𝑥 = 𝐶 𝑥 + 𝐷(𝑥) 𝐵(𝑥) where 𝐷(𝑥) 𝐵(𝑥) is a proper function. If 𝐶 𝑥 = 𝑐 where 𝑐 is a constant, then 𝐶 𝑥 is the horizontal asymptote. If 𝐶 𝑥 = 𝑐𝑥 + 𝑑, then 𝐶 𝑥 is the oblique asymptote.

Points & Lines of Symmetry SHALYN TAY COPYRIGHTED © A graph can be symmetric to the: Origin When rotated 180° about the origin, the same graph is obtained X-axis Equation stays the same when y is replaced with -y Y-axis Equation stays the same when x is replaced with -x Line 𝒚 = 𝒙 If (x,y) is on the graph, (y,x) is also on the graph. Line 𝒚 = −𝒙 If (x,y) is on the graph, (y,x) is also on the graph. 𝑦 = 𝑥 Graph is also symmetrical about the origin Cartesian Equations Expresses the relationship between x and y directly. Parametric Equations Expresses the relationship between x and y in terms of a parameter, t. You should be able to directly plot the graph from your G.C. by changing the mode to ’parametric’. Conversion from Parametric to Cartesian To convert equations from parametric to cartesian, we must eliminate the parameter using substitution. i.e. for 𝑥 = 𝑡2 + 2 and 𝑦 = 3𝑡 Cartesian equation: 𝑥 = ( 𝑦 3)2+2 𝑦 = −𝑥

Conics – Circles & Ellipses SHALYN TAY COPYRIGHTED © (ℎ, 𝑘) 𝑟 General Form 𝑥2 + 𝑦2 + 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 Standard Form (𝑥 − ℎ)2+(𝑦 − 𝑘)2= 𝑟2, where 𝑟 ≠ 0 Circles General Form 𝑏2𝑥2 + 𝑎2𝑦2 + 𝑐𝑥 + 𝑑𝑦 + 𝑒 = 0, where 𝑎 ≠ 0, 𝑏 ≠ 0, 𝑎 ≠ 𝑏 Standard Form (𝑥 − ℎ)2 𝑎2 + (𝑦 − 𝑘)2 𝑏2 = 1 Ellipses Circle Ellipse 𝑏 𝑏 𝑎𝑎 (ℎ, 𝑘) No Asymptotes

Conics – Hyperbolas SHALYN TAY COPYRIGHTED © General Form 𝑏2𝑥2 − 𝑎2𝑦2 + 𝑐𝑥 + 𝑑𝑦 + 𝑒 = 0, where 𝑎 ≠ 0, 𝑏 ≠ 0 Standard Form (𝑥 − ℎ)2 𝑎2 − (𝑦 − 𝑘)2 𝑏2 = 1 Left-Right Hyperbola General Form 𝑎2𝑦2 − 𝑏2𝑥2 + 𝑐𝑥 + 𝑑𝑦 + 𝑒 = 0, where 𝑎 ≠ 0, 𝑏 ≠ 0 Standard Form (𝑦 − 𝑘)2 𝑏2 − (𝑥 − ℎ)2 𝑎2 = 1 Top-Bottom Hyperbola Left-Right Hyperbola Top-Bottom Hyperbola Equation of Asymptotes: 𝑦 = ± 𝑏 𝑎 𝑥 − ℎ + 𝑘 (ℎ, 𝑘) (ℎ, 𝑘)

Conics – Special Hyperbolas SHALYN TAY COPYRIGHTED © Improper Form 𝑦 = 𝑎𝑥 + 𝑏 𝑐𝑥 + 𝑑 = 𝑙𝑖𝑛𝑒𝑎𝑟 𝑙𝑖𝑛𝑒𝑎𝑟 Proper Form 𝑦 = 𝑘 + 𝑚 𝑥−ℎ , where 𝑘 = 𝑎 𝑐 and h = − 𝑑 𝑐 Rectangular Hyperbola Hyperbola with an Oblique and Vertical Asymptote Rectangular Hyperbola Oblique-Vertical Asymptote Hyperbola Improper Form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 𝑑𝑥 + 𝑒 = 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑙𝑖𝑛𝑒𝑎𝑟 Proper Form 𝑦 = 𝑠𝑥 + 𝑡 + 𝑛 𝑥−ℎ , where h = − 𝑒 𝑑 Horizontal Asymptote: 𝑦 = 𝑘 = 𝑎 𝑐 Vertical Asymptote: x = ℎ = − 𝑑 𝑐 Oblique Asymptote: 𝑦 = 𝑠𝑥 + 𝑡 Vertical Asymptote: x = ℎ = − 𝑒 𝑑 n < 0 n > 0 m > 0 m < 0

TRANSFORMATION OF GRAPHS: TRANSLATION SCALING & REFLECTION MODULUS FUNCTIONS DERIVATIVE FUNCTIONS RECIPROCAL FUNCTIONS GRAPHING TECHNIQUES PART II SHALYN TAY COPYRIGHTED ©

Transformation of Graphs SHALYN TAY COPYRIGHTED © Translation in the direction of an axis is to move the graph in the direction of the axis (i.e. up, down, left or right) without changing its shape or size. Order of Transformations Replace x with (x - a) y = f (x – a) Translation of a units in the positive x direction Graph moves right Replace x with (x + a) y = f (x + a) Translation of a units in the negative x direction Graph moves left Replace y with (y - a) y = f (x) + a Translation of a units in the positive y direction Graph moves up Replace y with (y + a) y = f (x) - a Translation of a units in the negative y direction Graph moves down Scaling parallel to an axis changes the size of the curve, stretching it. Replace x with 𝑥 𝑎 y = f ( 𝑥 𝑎 ) Scaling parallel to the x-axis by a factor of a Remember to change the coordinates and asymptotes according to the new equationReplace y with 𝑦 𝑎 y = a f (x) Scaling parallel to the y-axis by a factor of a Reflection Replace x with -x y = f (-x) Reflection in the y-axis Flip left/right Replace y with -y y = - f (x) Reflection in the x-axis Flip up/down x y R Reflect T Translate S Scale R Reflect S Scale T Translate You may find the following acronym useful in helping you remember the order of transformations: Riley The Salesman Roasts Some Turkey