Connecting Transformational Geometry and Transformations of ...

Connecting Transformational Geometry and Transformations of Functions Introductory 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 only if there exists an isometry or sequence of isometries that maps one figure to another. The trivial rigid transformation is stated as the identity that maps a point (x, y) in the coordinate plane to itself; that is, I : (x, y)  (x, y) . Additionally, 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 necessarily distance. Similarities are created by composing an isometry 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. y x Figure 1: Translation of a point , to , The following notation represents this transformation , :

,

,

→

,



is the zero translation and is the identity.

Connection between Translations and Transformations of Functions

The graph in Figure 2 shows the translation of a parabola given by . 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 . y P' x 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 by , → , , if a point P has been translated to , 0, 5 and a For example, for =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 , 0 2, 5 1 2, 4 . In general, to find the relationship between the coordinates of a point and its image on the new graph, substitute x – a for x and y – b for y in the original function . Thus , right 2 units and 1 unit up, every For example, if we translate the graph of point on the translated graph satisfies the equation 2 1 and the point ′ 0, 5 lies on the The point 2, 4 lies on the graph of graph of a new function 2 1. Generally 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(x)

as So the translation of a function can be represented as follows: → . , : 2. Reflections over the x–axis and the y–axis Consider the reflection of a point with coordinates , over the x‐axis as shown in Figure 3a. The coordinates of the new point are , . Similarly the reflection of a point with coordinates , over the y‐axis, as shown in Figure 3(b), results in a point with coordinates , .

y y



x



x

(a): Reflection of (x, y) about the x‐axis (b): Reflection of (x, y) about the y‐axis Figure 3: Reflection of a point about different axes The following notation represents the reflection of the point , about the x‐axis and the reflection of the point , about the y‐axis, respectively. , → , : , → , : 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 , .



y





x

Figure 4: Rotation of a point (x, y) 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 fixed points. The number of fixed points helps characterize all isometries. The proof of this fact is accessible to advanced students. The characterization of isometries by the number of fixed points is represented by the following table: Isometries Fixed Points Identity

Entire plane is point‐wise fixed

Translation

0 fixed points or all points are fixed if it is a zero translation

Rotation

1 fixed point, the center of rotation,

or all points are fixed if it is a zero degree rotation Reflection about a line l

All points on the line l are fixed

Glide reflection

0 fixed points or if the translation involved is a zero translation, then there is a line of fixed points on the reflecting line.

The only non‐identity isometries that fix more than one point are reflections (or glide reflections with a zero translation). Thus, if a non‐identity isometry fixes two points, it must be a reflection that point‐wise fixes 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 axes is a rotation.

Reflections are generators of all isometries, and it suffices to use at most three reflections to get an arbitrary isometry. 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. Similarity Transformations Dilations The dilation of a point (x, y) to the point (ax, ay) where 0 is shown in Figure 5. The center of the dilation is the origin, and the scale factor is a.

y

x

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 , ,

on the graph of the original function

. Since

is the image of the point ,

lies on the graph of

, we have or

.

Composing a dilation with an isometry produces a similarity transformation. Consider an example 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 by the translation is given below. ,

For example,

,

:

→

2 ln

2

2

→

2

1 as seen in Figure 6.



y 5



2 ln

1

4

3



2



ln

1



-4

-3

-2

-1

O

-1

-2

-3

-4

1

2

3

4

x

2 ln

2



–5 2

1

Figure 6: Composition of dilation followed by a translation of f(x) = ln(x) Non‐Similarity Transformations The previous discussion centered on transformations typically studied in Euclidean geometry. This section concentrates on non‐similarity transformations to complete the connection to transformations of functions in an algebraic setting. First consider transformations that stretch in both the x–direction and the y–direction. Each of these stretches fixes a line. For a stretch in the x–direction (a horizontal stretch), the fixed line is the y–axis and for a vertical stretch the line is the x–axis. In general a stretch is often considered to be a composition of these two stretches. The stretches are specifically 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 x‐direction and in the y‐direction. Figure 7 illustrates a stretch of a point (x, y) to the point (ax, by) where 0 and 0. y



x



Figure 7: Stretch of the point (x, y) by a factor of a in the x–direction and a factor of b in the y–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 by . Consider the point ,

on the transformed graph. That point ,

the graph of the original function

. Since

,

is the image of the point lies on the graph of

,

on , we

have the following: or

.

Figure 8 illustrates the stretch transformation applied to the graph of sin . The graph is transformed by horizontally stretching by a factor of ½ and vertically stretching by a factor of 3. This transformation can be represented in two ways as follows: ,

:

sin

→

3 sin 2



or ,

,

→

,3





y 3

3sin 2 2

x

1

-4

-3

-2

-1

O

1

2

3

4

x

sin

-1

-2

-3



Figure 8: Stretch transformation of the sine graph



With a stretch transformation, similarity is lost if

.

a. Using the adopted notation above, the following can be expressed using stretches:

 S1,1

1.

The identity by I

2.

The reflection about the x‐axis by rx

 S1, 1

3.

The reflection about the y‐axis by ry

 S 1,1

4.

The dilation centered at the origin by a factor of a by DO,a

 Sa,a

5.

To stretch only horizontally by a factor of a, the transformation can be:

6. b.



S a ,1

To stretch only vertically by a factor of a, the transformation can be: S1, a

S a ,b 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 Finally, consider an example of the composition of a stretch and a translation. In general, a composition of these transformations can be represented as follows:

  x   xa  Ta,b  S h,k : y  f (x)   y  k f     y  k f    b  h h       



For example, consider the function

2

1

the composition of transformations of the function be written in the form shown below: 2

3 3

1. This function could be studied as √ . The transformed function can

1

Now the transformed function can be obtained from the original form composition of geometric transformations: T3,1  S3,2 .

√ by a

First the graph of 9.

√ is stretched and then the result is translated as shown in Figure

y

x

Figure 9: A composition of a stretch followed by a translation Conclusion

Students are rarely given the opportunity to study the connections between transformations in Euclidean geometry 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.