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5 Chapter Transforming functions Syllabus reference: 2.2, 2.3 cyan magenta yellow 95 100 50 Transformation of graphs Translations Stretches Reflections Miscellaneous transformations Simple rational functions The reciprocal of a function Modulus functions 75 25 95 A B C D E F G H 100 50 75 25 95 100 50 75 25 95 100 50 75 25 Contents: black Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\151IB_HL-3ed_05.cdr Thursday, 12 April 2012 12:24:59 PM BEN IB_HL-3ed 152 TRANSFORMING FUNCTIONS (Chapter 5) OPENING PROBLEM Consider the function f (x) = x3 ¡ 8, whose graph is shown alongside y x For each of the following functions g(x), draw the graphs of y = f (x) and y = g(x) on the same set of axes: a g(x) = x3 + b g(x) = (x ¡ 4)3 ¡ c g(x) = (¡x)3 ¡ d g(x) = ¡(x3 ¡ 8) ¡ ¢3 f g(x) = 12 x ¡ e g(x) = 2(x3 ¡ 8) ¯ ¯ g g(x) = ¯x3 ¡ 8¯ f(x) = x3 - h g(x) = jxj3 ¡ p j g(x) = x + i g(x) = x ¡8 GRAPHING PACKAGE Describe the transformation from y = f(x) to y = g(x) in each case A TRANSFORMATION OF GRAPHS There are several families of functions that we are already familiar with: Name General form Function notation Linear f (x) = ax + b, a 6= f : x 7! ax + b, a 6= Quadratic f (x) = ax + bx + c, a 6= f : x 7! ax2 + bx + c, a 6= Cubic f (x) = ax3 + bx2 + cx + d, a 6= f : x 7! ax3 + bx2 + cx + d, a 6= Modulus f (x) = jxj f : x 7! jxj Exponential f (x) = a , a > 0, a 6= f : x 7! ax , a > 0, a 6= Logarithmic f (x) = loge x or f (x) = ln x f : x 7! loge x or f : x 7! ln x Reciprocal f (x) = x k , x x 6= 0, k 6= f : x 7! k , x x 6= 0, k 6= These families of functions have different and distinctive graphs We can compare them by considering important graphical features such as axes intercepts, turning points, values of x for which the function does not exist, and asymptotes INVESTIGATION FUNCTION FAMILIES In this investigation you are encouraged to use the graphing package supplied Click on the icon to access this package GRAPHING PACKAGE What to do: cyan magenta yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\152IB_HL-3ed_05.cdr Monday, March 2012 4:57:22 PM BEN 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 From the menu, graph on the same set of axes: y = 2x + 1, y = 2x + 3, y = 2x ¡ Comment on all lines of the form y = 2x + b black IB_HL-3ed TRANSFORMING FUNCTIONS (Chapter 5) 153 From the menu, graph on the same set of axes: y = x + 2, y = 2x + 2, y = 4x + 2, y = ¡x + 2, y = ¡ 12 x + Comment on all lines of the form y = ax + On the same set of axes graph: y = x2 , y = 2x2 , y = 12 x2 , y = ¡x2 , y = ¡3x2 , y = ¡ 15 x2 Comment on all functions of the form y = ax2 , a 6= On the same set of axes graph: y = x2 , y = (x ¡ 1)2 + 2, y = (x + 1)2 ¡ 3, y = (x ¡ 2)2 ¡ and other functions of the form y = (x ¡ h)2 + k of your choice Comment on the functions of this form On the same set of axes, graph these functions: a y= , x y= , x c y= , x y= , x¡2 e y= , x y= + 2, x¡1 y= 10 x y= x+3 y= b y= ¡1 , x d y= , x y= y= ¡2 , x + 2, x y= ¡5 x y= ¡2 x ¡1 x+2 Write a brief report on your discoveries From the Investigation you should have observed how different parts of a function’s equation can affect its graph In particular, we can perform transformations of graphs to give the graph of a related function These transformations include translations, stretches, and reflections In this chapter we will consider transformations of the function y = f (x) into: ² y = f (x) + b, b is a constant ² y = f (x ¡ a), a is a constant ² y = pf(x), p is a positive constant ² y = f (qx), q is a positive constant ² y = ¡f (x) ² y = f (¡x) When we perform a transformation on a function, a point which does not move is called an invariant point Self Tutor Example If f (x) = x2 , find in simplest form: ³ ´ x a f (2x) b f ³ ´ x f ³ ´2 cyan magenta c d f(x) + = 2x + f(x + 3) ¡ = (x + 3)2 ¡ = x2 + 6x + ¡ yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\153IB_HL-3ed_05.cdr Monday, March 2012 4:57:49 PM BEN 95 100 50 = x2 + 6x + 75 x2 95 = 50 = 75 25 95 100 50 75 25 95 100 50 75 25 = 4x2 x 25 = (2x) d f(x + 3) ¡ 2 b f (2x) 100 a c 2f (x) + black IB_HL-3ed 154 TRANSFORMING FUNCTIONS (Chapter 5) EXERCISE 5A If f(x) = x, find in simplest form: a f (2x) b f (x) + 2 f (x) c If f(x) = x2 , find in simplest form: ³ ´ x a f (3x) b f d 2f (x) + c 3f(x) d 2f (x ¡ 1) + c d 2f (x + 1) ¡ 3 If f(x) = x3 , find in simplest form: a f (4x) b f (2x) f(x + 1) Hint: (x + 1)3 = x3 + 3x2 + 3x + See the binomial theorem in Chapter If f(x) = 2x , find in simplest form: a f (2x) If f(x) = b f (¡x) + 1 , x c f (x ¡ 2) + d 2f (x) + c 2f(x) + d 3f (x ¡ 1) + find in simplest form: b f ( 12 x) a f (¡x) B TRANSLATIONS INVESTIGATION TRANSLATIONS In this investigation we consider translations of the forms y = f(x) + b and y = f(x ¡ a) What to do: GRAPHING PACKAGE a For f (x) = x , find in simplest form: i f (x) + ii f(x) ¡ iii f (x) + b Graph all four functions on the same set of axes c What effect does the constant b have when y = f (x) is transformed to y = f (x) + b? a For f (x) = x2 , find in simplest form: i f (x ¡ 2) ii f (x + 1) iii f (x ¡ 5) b Graph all four functions on the same set of axes c What effect does the constant a have when y = f (x) is transformed to y = f (x ¡ a)? ² For y = f(x) + b, the effect of b is to translate the graph vertically through b units I I If b > it moves upwards If b < it moves downwards ² For y = f(x ¡ a), the effect of a is to translate the graph horizontally through a units I I If a > it moves to the right If a < it moves to the left cyan magenta yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\154IB_HL-3ed_05.cdr Monday, March 2012 4:58:15 PM BEN 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 ² For y = f(x ¡ a) + b, the graph is translated horizontally a units and vertically b units ¡ ¢ We say it is translated by the vector ab black IB_HL-3ed TRANSFORMING FUNCTIONS (Chapter 5) 155 EXERCISE 5B a Sketch the graph of y = x2 GRAPHING PACKAGE b On the same set of axes sketch the graphs of: i y = x2 + ii y = x2 ¡ c What is the connection between the graphs of y = f (x) and y = f (x) + b if: i b>0 ii b < 0? For each of the following functions f, sketch on the same set of axes the graphs of y = f (x), y = f (x) + 1, and y = f(x) ¡ a f(x) = 2x b f (x) = x3 d f(x) = (x ¡ 1)2 e f (x) = jxj c f (x) = x a On the same set of axes, graph f (x) = x2 , y = f (x ¡ 3), and y = f (x + 2) b What is the connection between the graphs of y = f(x) and y = f(x ¡ a) if: i a>0 ii a < 0? For each of the following functions f, sketch on the same set of axes the graphs of y = f (x), y = f (x ¡ 1), and y = f(x + 2) a f(x) = x3 b f (x) = ln x d f(x) = (x + 1)2 + e f (x) = jxj c f (x) = x For each of the following functions f, sketch on the same set of axes the graphs of y = f (x), y = f (x ¡ 2) + 3, and y = f(x + 1) ¡ a f(x) = x2 b f (x) = ex c f (x) = x Copy these functions and then draw the graph of y = f(x ¡ 2) ¡ y a y b y¡=¡f(x) y¡=¡f(x) -3 -2 x x -3 -2 The graph of f (x) = x2 ¡ 2x + is translated units right to g(x) Find g(x) in the form g(x) = ax2 + bx + c Suppose f (x) = x2 is transformed to g(x) = (x ¡ 3)2 + a Find the images of the following points on f (x) : i (0, 0) ii (¡3, 9) iii where x = b Find the points on f (x) which correspond to the following points on g(x) : cyan magenta iii (1 12 , 14 ) yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\155IB_HL-3ed_05.cdr Monday, March 2012 4:59:10 PM BEN 95 100 50 75 25 95 100 50 75 25 ii (¡2, 27) 95 100 50 75 25 95 100 50 75 25 i (1, 6) black IB_HL-3ed 156 TRANSFORMING FUNCTIONS (Chapter 5) C STRETCHES INVESTIGATION STRETCHES In this investigation we consider stretches of the forms y = pf(x), p > and y = f (qx), q > What to do: GRAPHING PACKAGE a For f (x) = x + 2, find in simplest form: i 3f (x) ii f (x) iii 5f (x) b Graph all four functions on the same set of axes c What effect does the constant p have when y = f(x) is transformed to y = pf(x), p > 0? a For f (x) = x2 , find in simplest form: i f (2x) ii f (3x) iii f ( x4 ) b Graph all four functions on the same set of axes c What effect does the constant q have when y = f (x) is transformed to y = f (qx), q > 0? ² For y = pf(x), p > 0, the effect of p is to vertically stretch the graph by the scale factor p I I If p > it moves points of y = f (x) further away from the x-axis If < p < it moves points of y = f (x) closer to the x-axis ² For y = f(qx), q > 0, the effect of q is to horizontally stretch the graph by the scale factor q I If q > it moves points of y = f (x) closer to the y-axis I If < q < it moves points of y = f (x) further away from the y-axis EXERCISE 5C Sketch, on the same set of axes, the graphs of y = f(x), y = 2f(x), and y = 3f(x) for each of: a f (x) = x2 b f(x) = x3 d f (x) = ln x e f(x) = c f (x) = ex x f f (x) = jxj Sketch, on the same set of axes, the graphs of y = f (x), y = 12 f(x), and y = 14 f (x) for each of: a f (x) = x2 b f(x) = x3 c f (x) = ex Sketch, on the same set of axes, the graphs of y = f (x) and y = f (2x) for each of: a y = x2 b y = (x ¡ 1)2 c y = (x + 3)2 Sketch, on the same set of axes, the graphs of y = f (x) and y = f ( x2 ) for each of: cyan c y = (x + 2)2 magenta yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\156IB_HL-3ed_05.cdr Monday, March 2012 4:59:39 PM BEN 95 100 50 75 25 95 100 50 75 25 95 100 50 75 b y = 2x 25 95 100 50 75 25 a y = x2 black IB_HL-3ed TRANSFORMING FUNCTIONS (Chapter 5) 157 Sketch, on the same set of axes, the graphs of y = f (x) and y = f(3x) for each of: b y = x2 a y=x c y = ex Consider the function f : x 7! x2 On the same set of axes sketch the graphs of: a y = f (x), y = 3f (x ¡ 2) + 1, and y = 2f (x + 1) ¡ b y = f (x), y = f (x ¡ 3), y = f( x2 ¡ 3), y = 2f( x2 ¡ 3), and y = 2f ( x2 ¡ 3) + c y = f (x) and y = 14 f (2x + 5) + a Given that the following points lie on y = f (x), find the coordinates of the point each moves to under the transformation y = 3f (2x) : i (3, ¡5) ii (1, 2) iii (¡2, 1) b Find the points on y = f (x) which are moved to the following points under the transformation y = 3f (2x) : i (2, 1) ii (¡3, 2) iii (¡7, 3) The function y = f (x) is transformed to the function y = + 2f ( 12 x + 1) a Fully describe the transformation that maps y = f (x) onto y = + 2f ( 12 x + 1) b Given that the following points lie on y = f (x), find the coordinates of the point each moves to under the transformation y = + 2f( 12 x + 1) i (1, ¡3) iii (¡1, ¡2) ii (2, 1) c Find the points on y = f (x) which are moved to the following points under the transformation y = + 2f ( 12 x + 1) i (¡2, ¡5) ii (1, ¡1) iii (5, 0) D REFLECTIONS INVESTIGATION REFLECTIONS In this investigation we consider reflections of the forms y = ¡f(x) and y = f(¡x) What to do: Consider f(x) = x3 ¡ GRAPHING PACKAGE a Find in simplest form: i ¡f (x) ii f(¡x) b Graph y = f(x), y = ¡f (x), and y = f (¡x) on the same set of axes Consider f(x) = ex a Find in simplest form: i ¡f (x) ii f(¡x) b Graph y = f(x), y = ¡f (x), and y = f (¡x) on the same set of axes cyan magenta yellow 95 100 50 75 25 95 b y = f (x) to y = f(¡x)? 100 50 75 25 95 100 50 75 25 95 100 50 75 25 What transformation moves: a y = f (x) to y = ¡f(x) black Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\157IB_HL-3ed_05.cdr Wednesday, 28 March 2012 4:42:27 PM BEN IB_HL-3ed 158 TRANSFORMING FUNCTIONS (Chapter 5) From the Investigation you should have discovered that: ² For y = ¡f (x), we reflect y = f (x) in the x-axis ² For y = f (¡x), we reflect y = f (x) in the y-axis In addition, in our earlier study of functions we found that: ² For y = f ¡1 (x), we reflect y = f(x) in the line y = x EXERCISE 5D On the same set of axes, sketch the graphs of y = f (x), y = ¡f (x), and if it exists, y = f ¡1 (x) a f (x) = 3x b f(x) = ex c f (x) = x2 d f (x) = ln x e f(x) = x3 ¡ f f (x) = 2(x + 1)2 For each of the following, find f (¡x) Hence graph y = f (x) and y = f (¡x) on the same set of axes a f (x) = 2x + b f (x) = x2 + 2x + c f(x) = x3 d f (x) = jx ¡ 3j The function f (x) = x3 ¡ ln x is reflected in the x-axis to g(x) Find g(x) The function f (x) = x4 ¡ 2x3 ¡ 3x2 + 5x ¡ is reflected in the y-axis to g(x) Find g(x) The function y = f (x) is transformed to g(x) = ¡f(x) a Find the points on g(x) corresponding to the following points on f (x): i (3, 0) ii (2, ¡1) iii (¡3, 2) b Find the points on f (x) that have been transformed to the following points on g(x): i (7, ¡1) ii (¡5, 0) iii (¡3, ¡2) The function y = f (x) is transformed to h(x) = f(¡x) a Find the image points on h(x) for the following points on f(x): i (2, ¡1) ii (0, 3) iii (¡1, 2) iv (3, 0) b Find the points on f (x) corresponding to the following points on h(x): i (5, ¡4) ii (0, 3) iii (2, 3) iv (3, 0) The function y = f (x) is transformed to m(x) = f ¡1 (x) a Find the image points on m(x) for the following points on f (x): i (3, 1) ii (¡2, 4) iii (0, ¡5) b Find the points on f (x) corresponding to the following points on m(x): i (¡1, 1) ii (6, 0) iii (3, ¡2) A function f (x) is transformed to the function g(x) = ¡f (¡x) a Describe the nature of the transformation b If (3, ¡7) lies on y = f(x), find the transformed point on y = g(x) cyan magenta 95 yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\158IB_HL-3ed_05.cdr Tuesday, 24 April 2012 9:38:00 AM BEN 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 c Find the point on f (x) that transforms to the point (¡5, ¡1) black IB_HL-3ed 159 TRANSFORMING FUNCTIONS (Chapter 5) y a Copy the graph of y = f(x) alongside, then draw the graph of: i y = ¡f(x) ii y = f (¡x) b Copy the graph of y = f (x) alongside y¡=¡f(x) i On the same set of axes, draw the reflection of y = f (x) in the line y = x ii Is this the graph of y = f ¡1 (x)? x -2 -2 E MISCELLANEOUS TRANSFORMATIONS A summary of all the transformations is given in the printable concept map CONCEPT MAP Self Tutor Example Consider f(x) = 2x + On separate sets of axes graph: a y = f (x) and y = f (x + 2) b y = f(x) and y = f(x) + c y = f (x) and y = 2f (x) d y = f(x) and y = ¡f (x) a b y y¡=¡f(x¡+¡2) y -2 -2 -2 y¡=¡f(x) O c +2 x -2 y x O O -2 95 yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\159IB_HL-3ed_05.cdr Thursday, March 2012 5:21:18 PM BEN 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 y¡=¡2f(x) magenta y¡=¡f(x) -1 cyan x O y¡=¡f(x) -2 y¡=¡f(x) d y +2 +2 +2 -2 -2 y¡= f(x)¡+¡2 black x y¡=¡-f(x) IB_HL-3ed 160 TRANSFORMING FUNCTIONS (Chapter 5) EXERCISE 5E Consider f (x) = x2 ¡ Invariant points not move under a transformation a Graph y = f (x) and state its axes intercepts b Graph the functions: i y = f(x) + iii y = 2f (x) ii y = f (x ¡ 1) iv y = ¡f (x) c What transformation on y = f(x) has occurred in each case in b? d On the same set of axes graph y = f(x) and y = ¡2f (x) Describe the transformation e What points on y = f (x) are invariant when y = f (x) is transformed to y = ¡2f (x)? On each of the following f (x) is mapped onto g(x) using a single transformation i Describe the transformation fully a ii Write g(x) in terms of f(x) b y y¡=¡f(x) y y¡=¡g(x) y¡=¡g(x) y¡=¡f(x) -2 -2 x x y c y d y¡=¡g(x) -4 x -1 y¡=¡g(x) x y¡=¡f(x) y¡=¡f(x) Copy the following graphs for y = f (x) and sketch the graphs of y = ¡f (x) on the same axes a b y c y y x x x Given the following graphs of y = f(x), sketch graphs of y = f(¡x): a b y c y y y=1 x x x cyan magenta 95 yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\160IB_HL-3ed_05.cdr Tuesday, 24 April 2012 9:39:01 AM BEN 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 x=2 black IB_HL-3ed 161 TRANSFORMING FUNCTIONS (Chapter 5) The scales on the graphs below are the same Match each equation to its graph A y = x4 a y B y = 2x4 C y = 12 x4 D y = 6x4 b c d y x y x y x For the graph of y = f(x) given, sketch the graph of: x y f(x) a y = 2f (x) b y= c y = f (x + 2) d y = f (2x) y¡=¡f(x) e y = f ( 12 x) y x For the graph of y = g(x) given, sketch the graph of: a y = g(x) + b y = ¡g(x) y = g(x) c y = g(¡x) d y = g(x + 1) -2 x y For the graph of y = h(x) given, sketch the graph of: y = h(x) b y = 12 h(x) ¡ ¢ d y = h x2 a y = h(x) + c y = h(¡x) -1 -2 F x (2, -2) SIMPLE RATIONAL FUNCTIONS In Chapter we introduced rational functions and some of their properties A function of the form y = ax + b , cx + d x 6= ¡ d c where a, b, c, and d are constants, is called a simple rational function cyan magenta yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\161IB_HL-3ed_05.cdr Monday, March 2012 5:09:49 PM BEN 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 These functions are characterised by the presence of both a horizontal asymptote and a vertical asymptote black IB_HL-3ed 162 TRANSFORMING FUNCTIONS (Chapter 5) y Any graph of a simple rational function can be obtained from the reciprocal function y = x by a combination of y= transformations including: x ² a translation (vertical and/or horizontal) ² stretches (vertical and/or horizontal) x For example: ² y= k x ² y= x¡k is a vertical stretch of y = x with scale factor k is a horizontal translation of y = x through k units Self Tutor Example is transformed by a vertical stretch with scale factor 2, The function g(x) results when y = x ¡ followed by a translation of ¡2 ¢ a Find an expression for g(x) b Find the asymptotes of y = g(x) c Sketch y = g(x) d Is g(x) a self-inverse function? Explain your answer a Under a vertical stretch with scale factor 2, f (x) becomes 2f (x) ³ ´ 1 ) becomes = x x x ¡ ¢ Under a translation of ¡32 , f(x) becomes f (x ¡ 3) ¡ 2 becomes x ) So, y = becomes x ¡ x¡3 g(x) = ¡2 x¡3 ¡ 2(x ¡ 3) = x¡3 ¡2x + = x¡3 b The asymptotes of y = are x = and y = x These are unchanged by the stretch, and shifted ¡ ¢ ¡2 by the translation ) the vertical asymptote is x = and the horizontal asymptote is y = ¡2 c y g(x) = d The graph is not symmetric about y = x, so g(x) is not a self-inverse function ¡2x + x¡3 x y = -2 cyan magenta yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\162IB_HL-3ed_05.cdr Monday, March 2012 5:13:27 PM BEN 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 x=3 black IB_HL-3ed TRANSFORMING FUNCTIONS (Chapter 5) 163 Self Tutor Example Consider the function f (x) = 2x ¡ x+1 a Find the asymptotes of y = f(x) b Discuss the behaviour of the graph near these asymptotes c Find the axes intercepts of y = f(x) d Sketch the graph of the function into y = f (x) x f Describe the transformations which transform y = f (x) into y = x e Describe the transformations which transform y = 2x ¡ x+1 2(x + 1) ¡ = x+1 ¡8 = +2 x+1 a f (x) = b As As As As y = f(x) is a translation of y = Now y = ¡ ¡1 ¢ ¡8 has asymptotes x = and y = x ) y = f (x) has vertical asymptote x = ¡1 and horizontal asymptote y = x ! ¡1¡ , y ! x ! ¡1+ , y ! ¡1 x ! ¡1, y ! 2+ x ! 1, y ! 2¡ c When x = 0, y = ¡8 through x d f (x) = 2x ¡ x+1 y y=2 ¡8 + = ¡6 ) the y-intercept is ¡6 x -6 When y = 0, 2x ¡ = ) x=3 ) the x-intercept is x = -1 becomes under a vertical stretch with scale factor x x ¡8 becomes under a reflection in the y-axis x x ¡ ¢ ¡8 ¡ becomes + under a translation through ¡21 x x+1 So, y = is transformed to y = f (x) under a vertical stretch with scale factor 8, followed x e by a reflection in the y-axis, followed by a translation through ¡ ¡1 ¢ f To transform y = f (x) into y = , we need to reverse the process in e x ¡ ¢ We need a translation through ¡12 , followed by a reflection in the y-axis, followed by a cyan magenta 95 yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\163IB_HL-3ed_05.cdr Monday, 30 April 2012 4:00:59 PM BEN 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 vertical stretch with scale factor 18 black IB_HL-3ed 164 TRANSFORMING FUNCTIONS (Chapter 5) EXERCISE 5F Write, in the form y = ax + b , the function that results when y = is transformed by: cx + d x a a vertical stretch with scale factor b a horizontal stretch with scale factor c a horizontal translation of ¡3 d a vertical translation of The function g(x) results when y = is transformed by a vertical stretch with scale factor 3, x ¡ ¢ followed by a translation of ¡1 a Write an expression for g(x) in the form g(x) = ax + b cx + d b Find the asymptotes of y = g(x) c State the domain and range of g(x) d Sketch y = g(x) e Is g(x) a self-inverse function? Explain your answer For each of the following functions f , find: i the asymptotes a f : x 7! ii how to transform y = 2x + x¡1 b f : x 7! into y = f (x) x 3x ¡ x+1 c f : x 7! 2x + 2¡x For each of the following functions f (x): i Find the asymptotes of y = f (x) ii Discuss the behaviour of the graph near these asymptotes iii Find the axes intercepts of y = f(x) iv Sketch the graph of y = f (x) into y = f (x) x vi Describe the transformations which transform y = f (x) into y = x v Describe the transformations which transform y = a y= 2x + x+1 b y= x¡2 c y= 2x ¡ 3¡x d y= 5x ¡ 2x + In order to remove noxious weeds from her property, Helga sprays with a weedicide The chemical is slow to act, and the number of weeds per hectare remaining after t days is modelled by N = 20 + 100 weeds/ha t+2 a How many weeds per were alive before the spraying? cyan magenta yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\164IB_HL-3ed_05.cdr Monday, March 2012 5:18:52 PM BEN 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 How many weeds will be alive after days? How long will it take for the number of weeds still alive to be 40/ha? Sketch the graph of N against t According to the model, is the spraying going to eradicate all weeds? 100 50 75 25 b c d e black IB_HL-3ed TRANSFORMING FUNCTIONS (Chapter 5) G 165 THE RECIPROCAL OF A FUNCTION f (x) For a function f (x), the reciprocal of the function is When y = is graphed from y = f (x): f (x) ² the zeros of y = f (x) become vertical asymptotes of y = f (x) ² the vertical asymptotes of y = f(x) become zeros of y = f (x) ² the local maxima of y = f (x) become local minima of y = f (x) ² the local minima of y = f(x) become local maxima of y = f (x) ² when f (x) > 0, 1 > and when f (x) < 0, is translated k units to the left, the resulting line is the same as when f (x) is translated ka units upwards The function f (x) results from transforming the function y = then a vertical stretch with scale factor 3, then a translation of by a reflection in the y-axis, x ¡1¢ a Find an expression for f(x) b Sketch y = f (x) and state its domain and range c Does y = f(x) have an inverse function? Explain your answer cyan magenta 95 yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\171IB_HL-3ed_05.cdr Tuesday, March 2012 9:53:06 AM BEN 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 d Is the function f a self-inverse function? Give graphical and algebraic evidence to support your answer black IB_HL-3ed 172 TRANSFORMING FUNCTIONS (Chapter 5) 10 Consider y = log4 x a Find the function which results from a translation of ¡ ¡2 ¢ b Sketch the original function and the translated function on the same set of axes c State the asymptotes of each function d State the domain and range of each function is transformed by a vertical stretch with scale factor x 11 The function g(x) results when y = 3, followed by a reflection in the y-axis, followed by a translation of units to the right a Write an expression for g(x) in the form g(x) = ax + b cx + d b Find the asymptotes of y = g(x) c State the domain and range of g(x) cyan magenta 95 yellow Y:\HAESE\IB_HL-3ed\IB_HL-3ed_05\172IB_HL-3ed_05.cdr Tuesday, March 2012 9:53:23 AM BEN 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 d Sketch y = g(x) black IB_HL-3ed ... the function does not exist, and asymptotes INVESTIGATION FUNCTION FAMILIES In this investigation you are encouraged to use the graphing package supplied Click on the icon to access this package... (¡x) B TRANSLATIONS INVESTIGATION TRANSLATIONS In this investigation we consider translations of the forms y = f(x) + b and y = f(x ¡ a) What to do: GRAPHING PACKAGE a For f (x) = x , find in simplest... (Chapter 5) C STRETCHES INVESTIGATION STRETCHES In this investigation we consider stretches of the forms y = pf(x), p > and y = f (qx), q > What to do: GRAPHING PACKAGE a For f (x) = x + 2, find