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Introduction

Introduction

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Introduction

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  1. 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. 5.1.2: Transformations As Functions

  2. Introduction, continued In 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. 5.1.2: Transformations As Functions

  3. Introduction, continued Once 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. 5.1.2: Transformations As Functions

  4. Introduction, continued First 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). 5.1.2: Transformations As Functions

  5. Introduction, continued Finally, 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. 5.1.2: Transformations As Functions

  6. Introduction, continued For example, if T(x, y) = (x + h, y + k), then would be: 5.1.2: Transformations As Functions

  7. 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. One-to-One Not One-to-One 5.1.2: Transformations As Functions

  8. 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). 5.1.2: Transformations As Functions

  9. 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. 5.1.2: Transformations As Functions

  10. 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. 5.1.2: Transformations As Functions

  11. Key Concepts, continued 5.1.2: Transformations As Functions

  12. Key Concepts, continued 5.1.2: Transformations As Functions

  13. 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. 5.1.2: Transformations As Functions

  14. Key Concepts, continued A A' B C B' C' Preimage Image 5.1.2: Transformations As Functions

  15. 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. 5.1.2: Transformations As Functions

  16. 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. 5.1.2: Transformations As Functions

  17. 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. 5.1.2: Transformations As Functions

  18. Key Concepts, continued If we have the dilation D(x, y) = (2x – 5, y – 4), we can graph as seen below. Note that . 5.1.2: Transformations As Functions

  19. Common Errors/Misconceptions taking functions in the wrong order not understanding that reflections, rotations, and translations have congruent preimages and images 5.1.2: Transformations As Functions

  20. Guided Practice Example 1 Given the point P(5, 3) and T(x, y)=(x + 2, y + 2), what are the coordinates of T(P)? 5.1.2: Transformations As Functions

  21. Guided Practice: Example 1, continued Identify the point given. We are givenP(5, 3). 5.1.2: Transformations As Functions

  22. Guided Practice: Example 1, continued Identify the transformation. We are given T(P)=(x + 2, y + 2). 5.1.2: Transformations As Functions

  23. Guided Practice: Example 1, continued Calculate the new coordinate. T(P)=(x + 2, y + 2) (5 + 2, 3 + 2) (7, 5) T(P) = (7, 5) ✔ 5.1.2: Transformations As Functions

  24. Guided Practice: Example 1, continued 5.1.2: Transformations As Functions

  25. Guided Practice Example 3 Given 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 . 5.1.2: Transformations As Functions

  26. Guided Practice: Example 3, continued Plot the points of the preimage. 5.1.2: Transformations As Functions

  27. Guided Practice: Example 3, continued Transform the points. T5, –3(x, y) = (x + 5, y – 3) 5.1.2: Transformations As Functions

  28. Guided Practice: Example 3, continued Plot the image points. 5.1.2: Transformations As Functions

  29. Guided Practice: Example 3, continued 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. ✔ 5.1.2: Transformations As Functions

  30. Guided Practice: Example 3, continued 5.1.2: Transformations As Functions

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