Introduction The word transform means “to change.” In geometry, a transformation changes the position, shape, or size of a figure on a coordinate plane. The original figure, called a preimage, is changed or moved, and the resulting figure is called an image. We will be focusing on three different transformations: translations, reflections, and rotations. These transformations are all examples of isometry, meaning the new image is congruent to the preimage. Figures are congruent if they both have the same shape, size, lines, and angles. The new image is simply moving to a new location. 1 5.1.2: Transformations As Functions
Transcript
IntroductionThe word transform means “to change.” In geometry, a transformation changes the position, shape, or size of a figure on a coordinate plane. The original figure, called a preimage, is changed or moved, and the resulting figure is called an image. We will be focusing on three different transformations: translations, reflections, and rotations. These transformations are all examples of isometry, meaning the new image is congruent to the preimage. Figures are congruent if they both have the same shape, size, lines, and angles. The new image is simply moving to a new location.
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5.1.2: Transformations As Functions
Introduction, continuedIn this lesson, we will learn to describe transformations as functions on points in the coordinate plane. Let’s first review functions and how they are written.
A function is a relationship between two sets of data, inputs and outputs, where the function of each input has exactly one output. Because of this relationship, functions are defined in terms of their potential inputs and outputs. For example, we can say that a function f takes real numbers as inputs and its outputs are also real numbers.
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5.1.2: Transformations As Functions
Introduction, continuedOnce we have determined what the potential inputs and outputs are for a given function, the next step is to define the exact relationship between the individual inputs and outputs. To do this, we need to have a name for each output in terms of the input. So, for a function f with input x, the output is called “f of x,” written f (x). For example, if we say f takes x as input and the output is x + 2, then we write f (x) = x + 2.
Now that we understand the idea of a function, we can discuss transformations in the coordinate plane as functions. 3
5.1.2: Transformations As Functions
Introduction, continuedFirst we need to determine our potential inputs and outputs. In the coordinate plane we define each coordinate, or point, in the form (x, y) where x and y are real numbers. This means we can describe the coordinate plane as the set of points of all real numbers x by all real numbers y. Therefore, the potential inputs for a transformation function f in the coordinate plane will be a real number coordinate pair, (x, y), and each output will be a real number coordinate pair, f (x, y). For example, f is a function in the coordinate plane such that f of f (x, y) is (x + 1, y + 2), which can be written as: f (x, y) = (x + 1, y + 2).
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5.1.2: Transformations As Functions
Introduction, continuedFinally, transformations are generally applied to a set of points such as a line, triangle, square, or other figure. In geometry, these figures are described by points, P, rather than coordinates (x, y), and transformation functions are often given the letters R, S, or T. Also, we will see T(x, y) written T(P) or P', known as “P prime.” Putting it all together, a transformation T on a point P is a function where T(P) is P'.
When a transformation is applied to a set of points, such as a triangle, then all points in the set are moved according to the transformation. 5
5.1.2: Transformations As Functions
Introduction, continuedFor example, if T(x, y) = (x + h, y + k), then would be:
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5.1.2: Transformations As Functions
Key Concepts• Transformations are one-to-one, which means each
point in the set of points will be mapped to exactly one other point and no other point will be mapped to that point.
• If a function is one-to-one, no elements are lost during the function.
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5.1.2: Transformations As Functions
Not One-to-OneOne-to-One
Key Concepts, continued• The simplest transformation is the identity function I
where I: (x', y' ) = (x, y).
• Transformations can be combined to form a new transformation that will be a new function.
• For example, if S(x, y) = (x + 3, y + 1) and T(x, y) = (x – 1, y + 2), then S(T(x, y)) = S((x – 1, y + 2)) = ((x – 1) + 3, (y + 2) + 1) = (x + 2, y + 3).
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Key Concepts, continued• It is important to understand that the order in which
functions are taken will affect the output. In the function on the previous slide, we see that S(T(x, y)) = (x + 2, y + 3). Does T(S(x, y) = (x + 2, y + 3)?
• In this case, T(S(x, y)) = S(T(x, y)), and in the graph at right we can see why.
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Key Concepts, continued
• However, here are two functions in where the order in
which they are taken changes the outcome: T2,3(x, y)
and a reflection through the line y = x, ry = x(x, y) on
. In the two graphs on the next slide,
we see and in the two graphs on
slide 12 we see . Notice the outcome
is different depending on the order of the functions.
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Key Concepts, continued
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5.1.2: Transformations As Functions
Key Concepts, continued
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5.1.2: Transformations As Functions
Key Concepts, continued• Because the order in which functions are taken can
affect the output, we always take functions in a specific order, working from the inside out. For example, if we are given the set of functions h(g(f(x))), we would take f(x) first and then g and finally h.
• Remember, an isometry is a transformation in which the preimage and the image are congruent. An isometry is also referred to as a “rigid transformation” because the shape still has the same size, area, angles, and line lengths. The previous example is an isometry because the image is congruent to the preimage. 13
5.1.2: Transformations As Functions
Key Concepts, continued
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5.1.2: Transformations As Functions
Preimage Image
A
B C
A'
B
'C
'
Key Concepts, continued• In this unit, we will be focusing on three isometric
transformations: translations, reflections, and rotations. A translation, or slide, is a transformation that moves each point of a figure the same distance in the same direction. A reflection, or flip, is a transformation where a mirror image is created. A rotation, or turn, is a transformation that turns a figure around a point.
• Some transformations are not isometric. Examples of non-isometric transformations are horizontal stretch and dilation. 15
5.1.2: Transformations As Functions
Key Concepts, continued• For example, a horizontal stretch transformation,
T(x, y) = (3x – 4, y), applied to is one-to-one—every point in is mapped to just one point in . However, horizontal distance is not preserved.
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Key Concepts, continued• Note that . From the graph,
we can see that and are not congruent; therefore T is not isometric.
• Another transformation that is not isometric is a dilation. A dilation stretches or contracts both coordinates.
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Key Concepts, continued• If we have the dilation D(x, y) = (2x – 5, y – 4), we
can graph , as seen below.
Note that . 18
5.1.2: Transformations As Functions
Common Errors/Misconceptions• taking functions in the wrong order
• not understanding that reflections, rotations, and translations have congruent preimages and images
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Guided Practice
Example 1Given the point P(5, 3) and T(x, y) = (x + 2, y + 2), what are the coordinates of T(P)?
Example 3Given the transformation of a translation T5, –3, and the points P (–2, 1) and Q (4, 1), show that the transformation of a translation is isometric by calculating the distances, or lengths, of and .
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Guided Practice: Example 3, continued
1. Plot the points of the preimage.
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5.1.2: Transformations As Functions
Guided Practice: Example 3, continued
2. Transform the points.T5, –3(x, y) = (x + 5, y – 3)
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Guided Practice: Example 3, continued
3. Plot the image points.
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Guided Practice: Example 3, continued
4. Calculate the distance, d, of each segment from the preimage and the image and compare them.Since the line segments are horizontal, count the number of units the segment spans to determine the distance.
d(PQ) = 5
The distances of the segments are the same. The translation of the segment is isometric.
