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# PRE Math 1B

UCI

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This 4 page Class Notes was uploaded by Adam Crona on Saturday September 12, 2015. The Class Notes belongs to Math 1B at University of California - Irvine taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/201873/math-1b-university-of-california-irvine in Mathematics (M) at University of California - Irvine.

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Date Created: 09/12/15

Jim Lambers IVIath 1B Fall Quarter 2004 05 Lecture 13 Notes These notes correspond to Section 57 in the text Transformations of Functions Mathematical models typically include relationships between quantities such position velocity or time and functions such exponential logarithmic or trigonometric functions are often ideal for describing such relationships Such functions that describe how a single quantity called the dependent variable depends on the value of a second quantity called the independent variable In most cases we will use the letter J to denote an independent variable or t if the independent variable is time The letter 9 will often be used to denote the corresponding dependent variable If fJ is the function that describes how 9 and J are related then we that y fJ that is the function f evaluated at a value J of the independent variable yields the corresponding value 9 for the dependent variable In the previous lectures we learned about the basic trigonometric functions and their graphs From these basic functions many other functions can be obtained by applying simple transforma tions which we now discuss o Shifting adding a constant to either the independent or the dependent variable causes the graph of a function to shift Adding a positive constant to the independent variable shifts the graph to the left while adding a negative constant shifts the graph to the right Adding a positive constant to the dependent variable shifts the graph upward while adding a negative constant shifts it downward Example 1 Figure 1 illustrates horizontal and vertical shifts of the function fJ sinJ E o Scaling Multiplying the independent or dependent variable by a constant has the effect of scaling the graph in some manner For instance multiplying the independent variable by a constant c where c gt 1 contracts the graph horizontally y a factor of 0 whereas if 0 lt c lt 1 the graph is stretched horizontally by a factor of 10 Similarly if the dependent variable is multiplied by a constant c where c gt 1 the graph is stretched vertically by a factor of 0 whereas it contracts vertically by a factor of 10 if 0 lt c lt 1 Example 2 Figure 2 illustrates horizontal and vertical scaling of the function sinJ Horizontal shi s of ysinx 1 2 r l 39 05 gt 0 1 o5r 1 l 0 1 2 3 4 5 6 7 Figure 1 Top plot horizontal shifts of sinx Note that adding a positive constant to J shifts the graph to the right while using a negative constant shifts to the left Bottom plot vertical shifts of sinx Note that adding a positive constant to the dependent variable shifts the graph up while a negative constant shifts the graph down Graphing y k A sinBx C and y k A cosBx 0 Many application areas deal with waves such sound waves radio waves ocean waves or light Such are described using functions in which the dependent variable continually oscillates within a small range of values in a cyclical pattern the independent variable changes Typically such functions have the form k A sinBJ C or k A cosBJ C where A B C and k are constants and B gt 0 Functions of this form are known simple harmonics and they are used to describe simple harmonic motion We already know from the previous lecture that if y sinJ or y cosJ then 9 oscillates indefinitely between 1 and 1 J changes and this oscillation occurs in a cycle that repeats itself every Err units since the sine and cosine functions are both Errperiodic However what if we wanted to use these functions to describe a wave that oscillated in a different pattern For Horizontal scaling of ysinx 7d Figure 2 Top plot horizontal scaling of sinx Note that scaling the independent variable by a constant greater than 1 contracts the graph horizontallv while using a positive constant less than 1 stretches it Bottom plot vertical scaling of sinx Scaling the dependent variable by a constant greater than 1 stretches the graph vertically while a positive constant less than 1 contracts it example the wave could oscillate between a different range of values or have a period of a different length In such cases we can use the graphical transformations discussed earlier in this lecture First we consider the graph of the curve 1 k A sinBJ C If we write this equation ykAsinB 1 then it is to describe the graph of this curve in terms of the graphical transformations we have discussed To obtain the graph of this curv we begin with the graph of y sin Then we scale the graph horizontally If B gt 1 then we contract the graph horizontally by a factor of 13 whereas if 0 lt B lt 1 then we stretch the graph horizontally by a factor of 113 The graph will now have a period of ErrB that is the graph will repeat lf every ErrB units in 1 instead of Err Next we shift the graph horizontally to the left by 0 units if CB gt 0 or to the right by CB units if CB lt 0 The number CB which represents the number of units that the graph is shifted to the right is called the phase shift of the curve Next we scale the graph vertically If gt 1 then we stretch the graph vertically by a factor of A whereas if 0 lt lt 1 then we contract the graph vertically by a factor of 1 In addition if A lt 0 the graph is also reflected about the The number Al is called the amplitude of the curve The t values of points on the curve deviate from the average t value by in either direction Finally we shift the graph vertically If k gt 0 then we shift the graph upward by k units whereas if k lt 0 then we shift the graph downward by M units Example 3 We will construct the graph of the simple harmonic y4 3cos21 6 2 from the graph of the simpler harmonic y cosJ using graphical transformations First we rewrite the equation 2 in the form 9 4 3cos2J 3 3 so that the phase shift can easily be seen to be 3 Then we perform the following transformations H first we contract the graph horizontally by a factor of 2 to obtain the graph of y cos21 N we shift the graph of y cos21 to the right by 3 units to obtain the graph of y cos2J 3 3 we stretch the graph vertically by a factor of 3 to obtain the graph of y 3 cos2J 3 4 we reflect the graph about the x axis to obtain the graph of t 3 cos2J 3 finally we shift the graph upward by 4 units to obtain the graph of y 4 3 cos2J 3 These transformations are shown in Figure 3 El

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