Mathematics
Function Transformations
Function transformations refer to the changes made to a function's graph by altering its equation. These changes include shifts, stretches, compressions, and reflections. These transformations can be applied to any type of function, including linear, quadratic, and trigonometric functions.
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7 Key excerpts on "Function Transformations"
No longer available |Learn moreFoundations of Mathematics
Algebra, Geometry, Trigonometry and Calculus
- Philip Brown(Author)
- 2016(Publication Date)
- Mercury Learning and Information(Publisher)
± 5.11 TRANSFORMATIONS OF FUNCTIONS Functions can be transformed in various ways. The transformations below result in a vertical or horizontal shift of the graph of the function, or a vertical or horizontal stretch or compression of the graph of the function. Graphs can also be reflected across the y axis or the x axis. 5.11.1 Vertical and Horizontal Shifts DEFINITION 5.11.1. Suppose that c > 0 and f(x) is any real-valued function, then the graphs of the following equations are related to the graph of = y f(x) by means of a shift up, down, left, or right. • y f x c = + ( ) (graph shifts up c units) • y f x c = − ( ) (graph shifts down c units) • y f x c = + ( ) (graph shifts to the left c units) • y f x c = − ( ) (graph shifts to the right c units) EXAMPLE 5.11.1. If f x x ( ) | | = and g x f x x ( ) ( ) | | = − = − 1 1 , then the effect of the transformation can be understood from table 5.4, which shows the values of f(x) and g(x) for some integer values of x. The graphs are shown in figure 5.18. ± - 3 - 2 - 1 1 2 3 x 1 2 3 y y = | x | y = | x - 1| FIGURE 5.18. The shifting of an absolute value graph. 144 • Foundations of Mathematics TABLE 5.4. Table of values for = = - y g x x ( ) | 1| x −3 −2 −1 0 1 2 3 = f x x ( ) | | 3 2 1 0 1 2 3 = - g x x ( ) | 1 | 4 3 2 1 0 1 2 5.11.2 Vertical and Horizontal Scaling DEFINITION 5.11.2. Suppose that c > 1 and f(x) is any real-valued function, then the graphs of the following equations are related to the graph of y f x = ( ) by means of a vertical or horizontal stretch- ing or compression of the graph. • y cf x = ( ) (graph stretches vertically) • = y f x c ( ) (graph compresses vertically) • y f cx = ( ) (graph compresses horizontally) • = y f x c (graph stretches horizontally) Examples of the scaling of cosine and sine graphs were given in section 4.8.3 There are more examples for other types of functions in the exercises at the end of this chapter. 5.11.3 Reflections Across the Axes DEFINITION 5.11.3.
eBook - PDF- Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
CHAPTER 9 New Functions from Old Overview Now we look at many ways of transforming or combining functions among all the families of functions we have studied. We start by transforming a function using stretching, com- pressing, shifting, or reflecting. The algebra of functions says that we can combine different functions through addition, subtraction, multiplication, or division. Certain power functions can be added to get polynomials, and polynomials can be divided to get rational functions. Composition of two functions is introduced and used to determine whether a function has an inverse. This final chapter closes with a collection of examples and Explore & Extends that weave together all the functions and techniques learned throughout your course. After reading this chapter, you should be able to • transform any function using stretches, compressions, shifting, and reflecting • combine any two functions using addition, subtraction, multiplication, and division • construct and interpret graphs of polynomials and their quotients, rational functions • compose two functions and look for inverse functions • use the families of functions and the algebraic tools introduced throughout the text to solve new problems 515 9.1 Transformations Finally, we look at the many ways we can create new functions from old, using examples from all the families of functions we have studied. This section considers transformations of func- tions and their graphs. What happens to the formula of a function when we shift, compress, or reflect the graph of the function? What happens to the graph of the function if we change the input or output of the function? Throughout the text, we have visualized the relationship between changes in the formulas of specific functions and changes in their graphs. Now we attach the language of functions to these changes and generalize to any function.
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 - 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- Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Graphing Functions by Using Vertical Stretching and Shrinking Translations and reflections are called rigid transformations because the basic shape of the curve being transformed is not changed. In other words, only the positions of the graphs have changed. Now we want to consider some transformations that distort the shape of the original figure somewhat. In Chapter 8, we graphed the equation y 5 2 x 2 by doubling the y coordinates of the or-dered pairs that satisfy the equation y 5 x 2 . We obtained a parabola with its vertex at the origin, symmetric with respect to the y axis, but narrower than the basic parabola. Likewise, we graphed the equation y 5 1 2 x 2 by halving the y coordinates of the ordered pairs that satisfy y 5 x 2 . In this case, we obtained a parabola with its vertex at the origin, symmetric with re-spect to the y axis, but wider than the basic parabola. We can use the concepts of “narrower” and “wider” to describe parabolas, but they can-not be used to describe other curves accurately. Instead, we use the more general concepts of vertical stretching and shrinking. Vertical Stretching and Shrinking The graph of y 5 cf ( x ) is obtained from the graph of y 5 f ( x ) by multiplying the y coordinates of y 5 f ( x ) by c . If 0 c 0 . 1 , the graph is said to be stretched by a factor of 0 c 0 , and if 0 , 0 c 0 , 1 , the graph is said to be shrunk by a factor of 0 c 0 . Graph f ( x ) 5 2 ! x and f ( x ) 5 1 2 ! x . Solution In Figure 9.23, the graph of f ( x ) 5 2 ! x is obtained by doubling the y coordinates of points on the graph of f ( x ) 5 ! x . Likewise, in Figure 9.23, the graph of f ( x ) 5 1 2 ! x is obtained by halving the y coordinates of points on the graph of f ( x ) 5 ! x . x f ( x ) = 2 x f ( x ) f ( x ) = x f ( x ) = x 1 2 Figure 9.23 Graphing Functions by Using Successive Transformations Some curves result from performing more than one transformation on a basic curve.
eBook - PDF- James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
This will give us a better understanding of how to graph functions. The transformations that we study are shifting, reflecting, and stretching. ■ Vertical Shifting Adding a constant to a function shifts its graph vertically: upward if the constant is positive and downward if it is negative. In general, suppose we know the graph of y f 1 x 2 . How do we obtain from it the graphs of y f 1 x 2 c and y f 1 x 2 c 1 c 0 2 The y -coordinate of each point on the graph of y f 1 x 2 c is c units above the y -coordinate of the corresponding point on the graph of y f 1 x 2 . So we obtain the graph of y f 1 x 2 c simply by shifting the graph of y f 1 x 2 upward c units. Similarly, we obtain the graph of y f 1 x 2 c by shifting the graph of y f 1 x 2 downward c units. VERTICAL SHIFTS OF GRAPHS Suppose c 0. To graph y f 1 x 2 c , shift the graph of y f 1 x 2 upward c units. To graph y f 1 x 2 c , shift the graph of y f 1 x 2 downward c units. c y x 0 c y x 0 y=f(x)+c y=f(x)-c y=f(x) y=f(x) EXAMPLE 1 ■ Vertical Shifts of Graphs Use the graph of f 1 x 2 x 2 to sketch the graph of each function. (a) g 1 x 2 x 2 3 (b) h 1 x 2 x 2 2 SOLUTION The function f 1 x 2 x 2 was graphed in Example 1(a), Section 2.2. It is sketched again in Figure 1. (a) Observe that g 1 x 2 x 2 3 f 1 x 2 3 So the y -coordinate of each point on the graph of g is 3 units above the corresponding point on the graph of f . This means that to graph g, we shift the graph of f upward 3 units, as in Figure 1. Recall that the graph of the function f is the same as the graph of the equation y f 1 x 2 . Copyright 2016 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.
eBook - PDFLinear Algebra
A First Course with Applications
- Larry E. Knop(Author)
- 2008(Publication Date)
- Chapman and Hall/CRC(Publisher)
C H A P T E R 5 Linear Transformations SECTION 5.1: TRANSFORMATION FUNDAMENTALS This isn ’ t Kansas anymore, Toto. Dorothy, in The Wizard of Oz We have found ourselves in some wild places and spaces and that is fi ne, as long as we can get home again. To tell the story of the spaces we have seen and the places we have been, we need ways to move from space to space and ways to compare spaces both in terms of the underlying sets and in terms of the operations on the sets. Functions (a.k.a. transformations in linear algebra) are the magic slippers that can take us where we want to go. Of course not every function has magic in its correspondences. Di ff erent functions have di ff erent proper-ties, and we need to identify those (magical) properties that can carry us home. * * * One way to compare two sets is by fi nding a pairing of the elements, if such a pairing is possible. For instance, the set a , b , c f g is comparable to the set 1, 2, 3 f g because the elements can be paired: a $ 1, b $ 2, and c $ 3. The set a , b , c f g , on the other hand, is not comparable to the set 1, 2, 3, 4 f g because there is no way to match the elements without either using some element from the fi rst set more than once or leaving some element in the second set unpaired. The pairing idea applies to both fi nite and in fi nite sets. For instance, the set of all real numbers R and the set of all positive real numbers R þ are comparable (in size) because the elements in the two sets can be completely paired. One such pairing is the correspondence x $ 2 x , which matches each real number with a corresponding positive real number. One property we require in a ‘‘ pairing ’’ is that di ff erent elements of A must match up with di ff erent elements of B. Mathematicians have a name for that property. De fi nition 1 : Let f : A ! B. The function f is one-to-one if and only if, for every x 1 and x 2 in A, if x 1 6 ¼ x 2 then f ( x 1 ) 6 ¼ f ( x 2 ). z 391
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