Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes. The concept of congruence can be introduced using isometries: Two geometric figures are congruent if and onl if there eists an isometr or sequence of isometries that maps one figure to another. The trivial rigid transformation is stated as the identit that maps a point (, ) in the coordinate plane to itself; that is, I :(, ) (, ). Additionall, when a transformation is applied to a point, the result is referred to as the image of that point under the particular transformation while the original point is the preimage of that image. Similarities are transformations that preserve shapes and angle measure but not necessaril distance. Similarities are created b composing an isometr and a dilation. Isometries 1. Translations First consider the translation of a point A with coordinates, to a point B with coordinates,. This translation moves the point A to the right a units and up b units if a and b are both positive as depicted in Figure 1. Figure 1: Translation of a point, to, The following notation represents this transformation, :,,, is the zero translation and is the identit. Connection between Translations and Transformations of Functions
The graph in Figure 2 shows the translation of a parabola given b. Consider the point on the translated graph (dotted graph). That point, is the image of the point, on the graph of the original function. Since the point, lies on the graph of, or. P' P Figure 2: Translation of the graph of The following discussion clarifies this derivation. Take a point P with coordinates, that lies on the dotted graph and look for the point on the original graph that maps to P ; the original point P has been moved a units to the right and b units up. That means that the point P on the original graph must have coordinates,. The relationship between the coordinates of P and P is given b,, For eample, for, if a point P has been translated to, 0, 5 and a =2 and b =1, that means move 2 units to the right and 1 unit up. The point P from the original function will be, 02,51 2,4. In general, to find the relationship between the coordinates of a point and its image on the new graph, substitute a for and b for in the original function. Thus For eample, if we translate the graph of, right 2 units and 1 unit up, ever point on the translated graph satisfies the equation 2 1 The point 2, 4 lies on the graph of and the point 0, 5 lies on the graph of a new function 2 1. Generall is translated to the left or the right, up or down, depending on the values of a and b to get the translation of f()
as So the translation of a function can be represented as follows:, :. 2. Reflections over the ais and the ais Consider the reflection of a point with coordinates, over the ais as shown in Figure 3a. The coordinates of the new point are,. Similarl the reflection of a point with coordinates, over the ais, as shown in Figure 3(b), results in a point with coordinates,. (a): Reflection of (, ) about the ais (b): Reflection of (, ) about the ais Figure 3: Reflection of a point about different aes The following notation represents the reflection of the point, about the ais and the reflection of the point, about the ais, respectivel. :,, :,, 3. Rotation about the origin (0,0) Consider a rotation about the point (0, 0) through an angle θ. In Figure 4, the image of the point, is,.
Figure 4: Rotation of a point (, ) to, about the origin through angle θ We use the following notation to represent the rotation of the point, about the origin,o, through an angle θ.,:,, cos sin,sin cos Notes: 1. A glide reflection is composed of a reflection and a translation parallel to the line of reflection. 2. Points that are their own image under transformations are fied points. The number of fied points helps characterize all isometries. The proof of this fact is accessible to advanced students. The characterization of isometries b the number of fied points is represented b the following table: Isometries Fied Points Identit Translation Rotation Entire plane is point wise fied 0 fied points or all points are fied if it is a zero translation 1 fied point, the center of rotation,
or all points are fied if it is a zero degree rotation Reflection about a line l Glide reflection All points on the line l are fied 0 fied points or if the translation involved is a zero translation, then there is a line of fied points on the reflecting line. The onl non identit isometries that fi more than one point are reflections (or glide reflections with a zero translation). Thus, if a non identit isometr fies two points, it must be a reflection that point wise fies the entire reflecting line. Some consequences of these are: a. The composition of two reflections about parallel lines is a translation. b. The composition of two reflections about intersecting aes is a rotation. Reflections are generators of all isometries, and it suffices to use at most three reflections to get an arbitrar isometr. A sketch of the proof of this statement is presented in the Geometric Transformations Sequence brief. 3. The composition of reflections, rotations, translations and glide reflections produces all isometries. Dilations Similarit Transformations The dilation of a point (, ) to the point (a, a) where 0 is shown in Figure 5. The center of the dilation is the origin, and the scale factor is a.
Figure 5: Dilation with origin as center and scale factor a This dilation is denoted as follows:, :,, Connection to Transformations of Functions Consider the point, on a dilated graph. That point, is the image of the point, on the graph of the original function. Since, lies on the graph of, we have or. Composing a dilation with an isometr produces a similarit transformation. Consider an eample of a composition of dilation and a translation. Consider the graph of ln under the dilation with center at the origin and scale factor 2 composed with a translation to the left 5 units and up 1 unit. The transformations do not commute and the transformation of the function under the composition of the dilation followed b the translation is given below.,, : 2 5 2 2 2 1 For eample, 2 ln 1 as seen in Figure 6. 5 4 2ln 1 3 2 1 ln -4-3 -2-1 O 1 2 3 4-1 -2 2ln 2-3 -4
Figure 6: Composition of dilation followed b a translation of f() = ln() Non Similarit Transformations The previous discussion centered on transformations tpicall studied in Euclidean geometr. This section concentrates on non similarit transformations to complete the connection to transformations of functions in an algebraic setting. First consider transformations that stretch in both the direction and the direction. Each of these stretches fies a line. For a stretch in the direction (a horizontal stretch), the fied line is the ais and for a vertical stretch the line is the ais. In general a stretch is often considered to be a composition of these two stretches. The stretches are specificall associated with stretch factors where the stretch factors can be greater than 1, smaller than 1 and negative. The word stretch is used here to indicate the composition of stretches in the direction and in the direction. Figure 7 illustrates a stretch of a point (, ) to the point (a, b) where 0 and 0. Figure 7: Stretch of the point (, ) b a factor of a in the direction and a factor of b in the direction This stretch transformation is denoted using the following notation:, :,,
Connection to Transformations of Functions The graph shown in Figure 7 illustrates the transformation of a point P. The same transformation could be applied to the graph of a function given b. Consider the point, on the transformed graph. That point, is the image of the point, on the graph of the original function. Since, lies on the graph of, we have the following: or. Figure 8 illustrates the stretch transformation applied to the graph of sin. The graph is transformed b horizontall stretching b a factor of ½ and verticall stretching b a factor of 3. This transformation can be represented in two was as follows:,: sin 3sin2 or,,, 3 3 2 3sin 2 1-4 -3-2 -1 O 1 2 3 4-1 sin -2-3 Figure 8: Stretch transformation of the sine graph
With a stretch transformation, similarit is lost if. a. Using the adopted notation above, the following can be epressed using stretches: 1. The identit b I S 1,1 r S 2. The reflection about the ais b 1, 1 r S 3. The reflection about the ais b 1,1 4. The dilation centered at the origin b a factor of a b D O,a S a,a 5. To stretch onl horizontall b a factor of a, the transformation can be: S a,1 6. To stretch onl verticall b a factor of a, the transformation can be: S 1,a b. Sab, preserves orientation if both a and b are positive or both a and b are negative. Otherwise, the orientation is reversed. Compositions of Stretches and Translations Finall, consider an eample of the composition of a stretch and a translation. In general, a composition of these transformations can be represented as follows: T a,b S h,k : f () kf kf h a h b For eample, consider the function 2 1 1. This function could be studied as the composition of transformations of the function. The transformed function can be written in the form shown below: 2 3 3 1 Now the transformed function can be obtained from the original form b a composition of geometric transformations: T 3,1 S 3,2.
First the graph of is stretched and then the result is translated as shown in Figure 9. Conclusion Figure 9: A composition of a stretch followed b a translation Students are rarel given the opportunit to stud the connections between transformations in Euclidean geometr and see their uses in graphing functions. If mathematics educators and mathematicians adopt common language and notation for use in both of these areas, it seems reasonable to conjecture that student learning would be enhanced. This is but one place where students often think that the two concepts are independent and unrelated to each other.