Mathematics
Transformations of Graphs
Transformations of graphs refer to the changes made to the shape, position, and size of a graph. These changes can be made by applying different mathematical operations such as translations, reflections, stretches, and compressions. Understanding these transformations is important in analyzing and interpreting graphs in various mathematical applications.
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7 Key excerpts on "Transformations of Graphs"
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Functions and Graphs 93 Thinking about graphs in terms of transformations is particularly valuable because it focuses attention on the role of the parameters in an equation and the resulting families of curves that result from varying those parameters, which in turn highlights the properties of the curves that remain constant and those that change. The examples discussed in this section have referred specifically to quadratic functions and circular functions, but the ideas obviously apply to the graphs of all functions. The ideas can be summarized succinctly for a general function y = /(*) as follows: • Translation of p units in the x direction and q units in the y direction: y^Ax)-+y=f(x-p) + q • Stretch by factor a in the x direction and factor b in the y direction: y=f(x)-*y = bf(?) However, most students find it easier to understand and remember key ideas by references to specific examples rather than in a more general form, so that a summary like that of Figure 6.15 is usually more appropriate. Figure 6.15 Summarizing the transformation of graphs CUBIC FUNCTIONS Another aspect of sketching graphs is how the graph of a function created by adding or subtracting a pair of functions is related to the two parent graphs. Figure 6.16 shows the graphs of y = jc 3 -x and y = x 3 -x 2 with the graph of y = x 3 shown dotted in each case. One of these would provide a useful focus for class discussion and then students could be asked to think out the other one for themselves. In the case of y = x 3 -x, it is clear that the curve must pass through the origin. When x is positive, a positive number is subtracted from x 3 at each point. It is as though the curve is pulled downwards, so that the graph lies below that of y = x 3 . Since x 3 -x = 0 when x -1, 94 Teaching and Learning Algebra the curve crosses the x axis at that point and is negative when x < 1 and positive when x > 1. This fully determines the position of the part of the curve on the positive side.
eBook - PDF- Ronald N. Umble, Zhigang Han(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 4 Translations, Rotations, and Reflections In grades 9–12 all students should apply transformations and use symmetry to...understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function nota-tion, and matrices . Principles and Standards for School Mathematics National Council of Teachers of Mathematics (2000) Studying geometric transformations provides opportunities for learners to describe patterns, discover basic features of isometries, make generalizations, and develop spatial competencies . H. Bahadir Yanick, Math. Educator Anadolu University In this chapter we consider three important families of isometries and investigate some of their properties. We motivate each section with an ex-ploratory activity. The instructions for these activities are generic and can be performed using any software package that supports geometric constructions. The Geometer’s Sketchpad commands required by these activities appear in the appendix at the end of the chapter. 4.1 Translations A translation of the plane is a transformation that slides the plane a finite distance in some direction. Exploratory activity 1, which follows below, uses the vector notation in the following definition: 45 46 Transformational Plane Geometry Definition 97 A vector v = a b is a quantity with norm (or magnitude) k v k := √ a 2 + b 2 and direction Θ defined by the equations k v k cos Θ = a and k v k sin Θ = b . The values a and b are called the x -component and y -component of v , respectively. The vector 0 = 0 0 , called the zero vector , has magnitude 0 and arbitrary direction. Given vectors v = a b and w = c d , the dot product is defined to be v · w := ac + bd . Thus v · v = a 2 + b 2 = k v k 2 . If P = x y and Q = x 0 y 0 are points, the vector PQ = x 0 -x y 0 -y is called the position vector from P to Q, and P and Q are called the initial and terminal points of PQ .
eBook - PDF- R. Gustafson, Jeff Hughes(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
One moment the most highly desired toy might be a high-performance vehicle racing down the highway, the next moment SOLUTION Self Check 7 EXAMPLE 7 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 319 Section 3.2 Transformations of the Graphs of Functions it might be a towering robot charging into battle. We are also familiar the science fiction action films Transformers, which are based on the Transformers toy. Nature also provides examples of transformation. Consider a butterfly. The four-stage development process of a butterfly is called metamorphosis, a Greek term meaning “transformation” or “change of shape.” ©Anyunov/Shutterstock.com ©KimPinPhotography/Shutterstock.com We can summarize the transformation ideas in this section as follows. Summary of Transformations If f is a function and k represents a positive number, then The graph of can be obtained by graphing y 5 f sxd and y 5 f sxd 1 k translating the graph k units upward. y 5 f sxd 2 k translating the graph k units downward. y 5 f sx 1 kd translating the graph k units to the left. y 5 f sx 2 kd translating the graph k units to the right. y 5 2f sxd reflecting the graph about the x-axis. y 5 f s2xd reflecting the graph about the y-axis. y 5 kf sxd k . 1 stretching the graph vertically by multi- plying each value of f sxd by k. y 5 kf sxd 0 , k , 1 shrinking the graph vertically by multi- plying each value f sxd by k. y 5 f skxd k . 1 shrinking the graph horizontally by mul- tiplying each x-value of f sxd by 1 k .
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Figure 16.1 Notice that when we fold the kite over BD, we are actually forming a one-to-one correspondence between the points of the kite. For example, points A and C cor- respond to each other, points along segments AB and CB correspond, and points along segments AD and CD correspond. In this chapter we will investigate correspon- dences between points of the plane. A transformation is a one-to-one correspondence between points in the plane such that each point P is associated with a unique point ′ P , called the image of P. Transformations that preserve the size and shape of geometric figures are called isometries (iso means “same” and metry means “measure”) or rigid motions. In the remainder of this subsection, we’ll study the various types of isometries. Translations Consider the following transformation that acts like a “slide.” NCTM Standard All students should recognize and apply slides, flips, and turns. Children’s Literature www.wiley.com/college/musser See “A Cloak for the Dreamer” by Aileen Friedman. Reflection from Research Learning about transformations in a computer-based environment significantly increases students‘ two-dimensional visualization (Dixon, 1995). Common Core – Grade 8 Verify experimentally the proper- ties of translations that lines are taken to lines, and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are take to parallel lines. In each of the following three tilings, describe the type of motion required to move Figure 1 to Figure 2 which will move points A to ′ A , B to ′ B , C to ′ C , and D to ′ D . A 1 B C D A9 2 D9 C9 B9 A 1 B C D A9 2 B9 C9 D9 A 1 B C D A9 2 B9 C9 D9 TRANSFORMATIONS 824 Chapter 16 Geometry Using Transformations The sliding motion of Example 16.1 can be described by specifying the distance and direction of the slide. The arrow from A to ′ A in Figure 16.3 conveys this infor- mation.
- Ronald Goldman(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
Chapter 4 Af fi ne Transformations for Satan himself is transformed into an angel of light. – 2 Corinthians 11:14 4.1 Transformations Transformations are the lifeblood of geometry. Euclidean geometry is based on rigid motions — translation and rotation — transformations that preserve distances and angles. Congruent triangles are triangles where corresponding lengths and angles match. Transformations generate geometry. The turtle uses translation (FORWARD), rotation (TURN), and uniform scaling (RESIZE) to generate curves by moving about on the plane. We can also apply translation (SHIFT), rotation (SPIN), and uniform scaling (SCALE) to build new shapes from previously de fi ned turtle programs. These three transformations — translation, rotation, and uniform scaling — are called conformal transformations . Conformal transformations preserve angles, but not distances. Similar triangles are triangles where corresponding angles agree, but the lengths of corresponding sides are scaled. The ability to scale is what allows the turtle to generate self-similar fractals like the Sierpinski gasket. In Computer Graphics, transformations are employed to position, orient, and scale objects as well as to model shape. Much of elementary Computational Geometry and Computer Graphics is based upon an understanding of the effects of different fundamental transformations. The transformations that appear most often in two-dimensional Computer Graphics are the af fi ne transformations. Af fi ne transformations are composites of four basic types of transformations: translation, rotation, scaling (uniform and nonuniform), and shear. Af fi ne transformations do not necessarily preserve either distances or angles, but af fi ne transformations map straight lines to straight lines and af fi ne transformations preserve ratios of distances along straight lines (see Figure 4.1). For example, af fi ne transformations map midpoints to midpoints.
eBook - PDF- David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 3.4 Techniques in Graphing 169 EXAMPLE SOLUTION 2 Applying Reflection and Translations to Use transformations of the graph of to graph the equation . Start with the equation y x and its graph. See Figure 6. Replace x by x 2 to shift the graph of to the left 2 units and obtain the equation y 0 x 2 0 y 0 x 0 y 0 x 2 0 3 y 0 x 0 y x Now let’s look at reflection of a graph in the x -axis. Consider the graph of . Replace y by y to obtain or . For each x 0 the graph of the new equation has a y -coordinate opposite in sign to that on the original graph. See Figure 5(a). So the graph of can be obtained from the graph of by reflecting the graph of in the x -axis. y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x PROPERTY SUMMARY Reflections in the x - and y -axes Let f be a function. Replacing y by y in the equation y f ( x ) gives the new equation y f ( x ) or y f ( x ) whose graph is that of y f ( x ) reflected in the x -axis. Similarly, replacing x by x in the equation y f ( x ) gives the new equation y f ( x ) whose graph is that of y f ( x ) reflected in the y -axis. (a) x y (1, 1) (1, _1) y=œ „ x y=_œ „ x (b) „ x y (1, 1) (_1, 1) y=œ „ x y=œ „ _x To g raph y=_œ„ x , reflect the g raph of y=œ„ x in the x -axis. More g enerally, to g raph y=_ƒ , reflect the g raph of y=ƒ in the x -axis. To g raph y=œ„ _x , reflect the g raph of y=œ„ x in the y -axis. More g enerally, to g raph y=f(_x) , reflect the g raph of y=ƒ in the y -axis. Figure 5 Consider the graph of , where x 0. If we replace x by x we get a new equation , where x 0 or x 0.
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
EXAMPLE 1 Identifying Transformations in Art What types of transformations are shown in the tessellation? Explain your reasoning. SOLUTION There are translations and rotations. Rotation Translation There are no reflections or dilations in this tessellation. For the tessellation shown in Example 1, you can convince your students that there are no dilations because all of the lizards are congruent. To convince your students that there are no reflections, make a copy of the print, cut out one of the lizards, and reflect it. The reflected image does not occur in the print. Reflection Classroom Tip Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 532 Chapter 14 Transformations Translations Definition of Translation A translation is a transformation in which a figure slides but does not turn. Every point of the figure moves the same distance and in the same direction. Slide Slide The original figure and its image are congruent. EXAMPLE 2 Identifying Translations Determine whether the blue figure is a translation of the red figure. a. b. SOLUTION a. The red trapezoid slides to form the blue trapezoid. So, the blue trapezoid is a translation of the red trapezoid. b. The red 3 turns to form the blue 3. So, the blue 3 is not a translation of the red 3. EXAMPLE 3 Identifying Translations in Real Life You rearrange the icons on your smartphone. Do the icons represent examples of translations? Explain your reasoning. Original Rearranged SOLUTION. Except for the message icon ( ) , all of the icons slide without changing their size or shape.
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