4 GRAPH TRANSFORMS Objectives After studing this chapter ou should be able to use appropriate technolog to investigate graphical transformations; understand how complicated functions can be built up from transformations of simple functions; be able to predict the graph of functions after various transformations. 4.0 Introduction You have alread seen that the abilit to illustrate a function graphicall is a ver useful one. Graphs can easil be used to eplain or predict, so it is important to be able to sketch quickl the main features of a graph of a function. New technolog, particularl graphic calculators, provides ver useful tools for finding shapes but as a mathematician ou will still need to gain the abilit to understand what effect various transformations have on the graph of a function. First tr the activit below without using a graphic calculator or computer. Activit You should be familiar with the graph of =. It is shown on the right. Without using an detailed calculations or technolog, predict the shape of the graphs of the following = (a) = + (b) = (c) = 3 (d) = ( ) (e) = ( +) (f) = ( 0). Now check our answers using a graphic calculator or computer. 49
4. Transformation of aes Suppose =, then the graph of = + moves the curve up b two units as is shown in the figure opposite. For an value, the value will be increased b two units. = + = What does the graph of = + a look like? What ou are doing in the eample above is equivalent to moving the aes down b units, which ou can see b defining Y =. Then Y = and ou are back to the original equation. Describe the graph of = 3 + This tpe of transformation f () a f () + a is called a translation of the graph b a units along the -ais. Eample Find the value of a so that = +a just touches the -ais. = The graph of = is shown opposite. From this, ou can see that it needs to be raised one unit, since its minimum value of is obtained at =. So the new equation will be 0 Y = +. Note that the new Y function can be written as - Y = ( ) 0 for all and equalit onl occurs when = (as illustrated). As well as translations along the -ais, ou can perform similar operations along the -ais. Y Y = ( ) 0 50
Activit Translations parallel to the -ais Again use the familiar = curve, but this time write it as f () =. Evaluate f ( ). Sketch the graph of = f ( ). What is the relationship between this curve and the original. If ou know the shape of = f( ), what does the graph of = f ( a) look like? The transformation f () a f ( a) is a translation b a units along the -ais. Eample The function f () is defined b f () = 3 3 +3. B considering = f ( +), deduce the shape of the graph of f( ). f( +)= ( +) 3 3( +) +3( +) ( +) = ( + )( + ) = + + ( + ) 3 = ( + )( + ) = ( + )( + + ) = 3 + 3 + 3 + f ( +) = 3 + 3 + 3 + 3( ++) + 3( + ) = 3 + (+3 3)+(3 6 + 3) + 3+3 = 3 Hence = f ( +) = 3 and this is illustrated opposite. = 3 0 This means that f( ) must also have this shape, but moved one unit along the -ais. = 3 3 +3 5
Activit 3 Using a graphic calculator or computer, (a) illustrate the curves = 3 and = 3 3 +3 and hence verif the result in the sketch on the previous page; (b) illustrate the curves = 4 and = 4 8 3 + 4 3 +6 and deduce a simpler form to write the second function. Use range to 4 and range 0 to 0. Eercise 4A. Without using a graph plotting device, draw sketches of f (), f ( + 5), f () + 5 for the following functions (a) f () = (b) f () = (c) f () = ( ).. Use a graph plotting device to illustrate the graphs of f () =, g()= ++. Hence or otherwise write g() in the form f ( + a) + b b finding the constants a and b. 3. If f () =, sketch the graphs of (a) f () (b) f ( ) (c) f ( ) +. 4. Stretches In this section ou will be investigating the effect of stretching either the - or -ais. Eample For the function draw the graphs of 5 = f () = + (a) f () (b) f ( ) (c) f () (d) f (). (a) f () = + (b) f ( ) = + these are illustrated opposite 0 = + = + = +
(c) f () = ( +) (d) f () = ( +) again illustrated opposite You should be beginning to get a feel for what the various tpes of transformations do, and the net activit will help ou to clarif our ideas. = ( +) = + = ( +) Activit 4 Stretches - 0 For the function draw the graphs of = f () = + (a) f () (b) f () (c) f () (d) f ( ). Use a graph plotting device to help ou if ou are not sure of what the graphs look like. The eample and the activit have shown ou that = α f () is a stretch, parallel to the -ais, b a factor α = f (α ) is a stretch, parallel to the -ais, b a factor α Eample If f () =, illustrate (a) f () (b) f ( ). = (a) f () = ; this is illustrated opposite. = (b) f ( ) = ( ) =, which is identical to f (). For this rather special function, a stretch of factor α parallel to the -ais is identical to a stretch of factor α parallel to the - ais. Wh are the two transformations identical for the function =? 53
Eercise 4B. For the function f () = illustrate the graphs of (a) f ( ) (b) f () (c) f () (d) f ().. For which of the following does the function = f () remain unaltered b the transformation 3. For the function = f (), shown below, sketch the curves defined b (a) = f ( ) (b) = f (). = α f (α )? (a) f () = (b) f () = + 0 - π π (c) f () = (d) f () =. 4.3 Reflections If f () = +, then the graph of f () = (+) is seen to be a reflection in the -ais. = f ( ) = + On the other hand f ( ) = + can be seen to be a reflection in the -ais. Activit 5 Reflections = f() For each of the functions below sketch (i) f () (ii) f ( ): (a) f () = (b) f () = + (c) f () = 3 (d) f () =. Use a graphic calculator or computer to check our answers if ou have an doubt. You have seen that f ( )is a reflection in the -ais f( ) is a reflection in the -ais It is now possible to combine various transformations. Eample If the graph of = f () is shown opposite, illustrate the shape of = f ( ) + 3. 54 - - - 3
To find f ( ), ou reflect in the -ais to give the graph opposite. - - - 3 The sketch of f ( ) is shown opposite. This is a stretch of factor along the -ais. - - - 3 4-5 4 Finall adding 3 to each value gives the graph shown opposite, 3 = f ( ) + 3. - - 0 3 4 5 Eample Sketch the graph of = +. Of course, ou could find its graph ver quickl using a graphic calculator or computer. It is, though, instructive to build up the sketch starting from a simple function, sa f () = and performing transformations to obtain the required function. In terms of f, ou can write = f ( ) +. So ou must first sketch = f ( ) as shown opposite. 55
Now sketch = f ( ) as shown opposite. Finall ou add to the function to give the sketch opposite. The abilit to sketch curves quickl will be ver useful throughout our course of stud of mathematics. Although modern technolog does make it much easier to find graphs, the process of understanding both what various transformations do and how more comple functions can be built up from a simple function is crucial for becoming a competent mathematician. Eercise 4C. The graph below is a sketch of = f (), showing three points A, B and C. A Sketch a graph of the following functions: B C. Using the functions f () =, g() = show how each of the following functions can be epressed in terms of f or g. Hence sketch these graphs. (a) = + (b) = 4 (c) = ( + 4) + ( 4) (d) = + ( 0) (e) = + + 4. (a) f ( ) (b) f () (c) f 3 (d) f () (e) f ( + 3) (f) f ( +) (g) f () + 5. In each case indicate the position of A, B and C on the transformed graphs. 56
4.4 Miscellaneous Eercises Chapter 4 Graph Transforms. The function = f () is illustrated below.. Epress each of the following functions in terms of either f () = or g() =. (a) () = 4 + (b) () = ( +) ( ). - 3. Sketch the graph of Sketch the following functions: (a) = f() (b) = f ( ) (c) = f () + (d) = f (). f () = 3 ( 0). Show that f ( ) = f(). What does this tell ou about the function? 57
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