Objective: Use calculator to comprehend transformations.

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1 math111 (Bradford) Worksheet #1 Due Date: Objective: Use calculator to comprehend transformations. Here is a warm up for exploring manipulations of functions. specific formula for a function, say, Given a f(x) = 2 x 2 8 x + 5, write the formulas for the following: f(x) + 1, f(x + 1), f(x), f( x), f(2 x), 2 f(x). There is more room on the next page. The answers are are at the end of the worksheet. 1

2 Check your answers with those at the end of the worksheet. 2

3 In this worksheet, you play the role of an empirical scientist. You have three major goals to accomplish. They are listed here: Supply the calculator with the correct data and settings. Observe and record the effects of transformations on various functions. Generalize specific observations to make correct predictions about the effects of transformations on functions. For this worksheet you will perform transformations on a periodic wave function f(x) = sin(x). The name of the function is SIN, which is pronounced sine. Here is a computer drawn graph of the sine function. y 2 y = sin(x) 4 8 x. 2 Figure 1: Graph of sine wave 3

4 Two particularly interesting functions to use for studying transformations are y = sin(x) and y = cos(x). You do not need to worry about the significance of these functions. You will study them later. Try to discover the patterns in the graphs. Your graphing calculator acts as a very fast point plotter, which shows the graphs of the functions. To prepare yourself, use your calculator to graph f(x) = sin(x). Notice your x and y scales. Radian mode is required. Sketch the graph here. f(x) = sin(x) Shifts The first basic kind of transformation is adding a real number to the output. You can add a positive real number to sin(x). Also, you can subtract a positive real number from sin(x). On your calculator graph f(x) = sin(x) and y = sin(x) + 3. Next, overlay the graph y = sin(x) 2. Describe, in general terms, what happens to the graph of a function when you add a positive constant to the function. 4

5 A new kind of change you can make is add to the input of a function, i.e. f(x + k). By changing the input, you are affecting the independent variable. The input is altered and then the function is performed. What happens to the original graph? What do you expect the graph of y = sin(x + k) to look like? Make your prediction. Now graph f(x) = sin(x) and overlay the graph of g(x) = f(x + k) = sin(x + k). (You choose a value for k. Avoid the value k = 6.) Is this what you expected? Explain what happened and why. 5

6 By now you should be able to look at the formula for a function and tell whether its graph involves any basic shifts. You should be able to predict the magnitudes and directions of these shifts. Try these: predict the kind of shift, then have the graphing calculator draw the graph. sin(x + π) prediction: cos(x 2) prediction: 1 + sin(x 3) prediction: cos(x π/2) + 3 prediction: 6

7 Reflections Another way to transform graphs is by reflections. On your calculator graph y = sin(x) and y = sin(x). With the graphs of sin(x) and sin(x) on the screen describe the relationship between the two. In the search for patterns, you might expect a similar relationship between any pair f(x) and f(x), that is, a function and its opposite. Try graphing these functions and their opposites. f(x) = cos(x) and g(x) = f(x) y y = cos(x) x. 7

8 f(x) = sin(x) + 2 and g(x) = f(x) f(x) = cos(x) π and g(x) = f(x) 8

9 You have been comparing a function to its opposite. Now consider the effect of a reflection of the independent variable, namely f( x). Try this on the function f(x) = sin(x). Draw the graph of y = sin( x). Does this graph look familiar? Can you establish that f( x) = f(x) for f(x) = sin(x)? (Yes or No) (A function where f( x) = f(x) is called an odd function.) 9

10 Vertical Stretching and Compression c f(x) In this part of the worksheet, you ll look at another way to modify a function and watch what happens to its graph. On your calculator graph sin(x) on the interval 10 x 10. Overlay the graph of 3 sin(x). Next overlay the graph of 1 sin(x). Draw the graphs here. 2 Describe what the 3 does to the graph of y = sin(x). 10

11 Describe what the 1 does to the graph of y = sin(x). 2 Horizontal Stretching and Compression f(c x) Another way to modify a function is by an inside change. Suppose you multiply the independent variable by a constant. The effect on the graph is predictable. On the calculator, you see that the functions sin(x) and cos(x) reveal the inside change quite well. These functions show the difference between multiplying the function and multiplying the independent variable. Recall, from the previous section, what happened to the graph of sin(x) when you changed it to 3 sin(x). Show by means of a sketch that the 2 in sin(2 x) compresses the graph of sin(x) horizontally by a factor of 1/2. Graph on next page. 11

12 What would you expect of sin ( 1 2 x)? On your calculator Graph sin(x) and sin ( 1 2 x). Describe what you see. Will the function f(x) = cos(x) behave in the same manner? Choose a value for c (something other than 2); on your calculator graph c cos(x), and cos(c x) 12

13 Ultimate Challenge: What is the formula for the function f(x) = sin(x) shifted π to the left, then stretched vertically by a factor of 3, then compressed horizontally by a factor of 1/6, and then shifted up 5? Check the 2 answer with your graphing utility. Answers to preparation exercises: If f(x) is 2 x 2 8 x + 5, then f(x) + 1 is 2 x 2 8 x + 6 f(x + 1) is 2 (x + 1) 2 8 (x + 1) + 5 = 2 x 2 4 x 1 f(x) is 2 x x 5 f( x) is 2 x x + 5 f(2x) is 8 x 2 16 x f(x) is 4 x 2 16 x Done in TEX.

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