Lab 6: Some Important Transformations for Trig Functions 1. You worked with transformations of functions earlier. For example. if you started withy = 1:2 what would happen when you graphed y = [x 3)2 ? a. What do you think the graph of y = sin (2: + 5) will look like compared to the graph of y = sin x? Discuss this with your group and make a conjecture. b. Use Desmos to graph 3; = sinx and y = sin (x + E). What do you notice? 2. Now think about 1? = cosx and y = cos (x + E). a. What do you anticipate will happen? h. Graph both to check your prediction. c. How does the new graph compare to y = sinx? 3. What would happen if you graphed y = tan(x + E)? Explain how you could you know this without graphing. 4. Try graphing y = cos (x). a. Describe what happened. Why? h. Will the same thing happen when you graph 1? = sin{x)? c. What happened? d. How could you describe your new graph in terms of sin x? 5. What will the graph of J? = sinx look like? Sketch it. 6. Based on your earlier work with transformations of functions. without Demos. describe the graph of y = 3 sin 2:. Then use Desmos to check your conjecture. ?. Now graph use Desmos and sketch a graph of y = sin 2x. Explain the effect of the "2" in this equation and how the graph is related to the graph of y = sinx. 3. Match the following equations with their graphs and write an explanation of their relationships to the parent function y = sinx and its graph. There's one extra equation. y = 4 sin x 2 y = sin 4x 1 y = sin - x -2TT 2TT 1 y = - sin x y = sin(-4x) y = sin x - 4 -2 AAAAAAAAAAA -2 -5 2 0 2TT 2TT -2EXTRA The sine, cosine, and tangent are the important trigonometric functions to know and use. There are three other functions based on them, the secant, cosecant, and cotangent. They are defined as follows: sec 0 = - CSC 8 = cot 0 = coso cos 0 sing sing 9. Use Desmos to graph y = sec x and sketch it in your notebook. Be sure you put Desmos in radian mode. a. Label the asymptotes on your graph using dotted lines. b. Using Desmos, sketch the graph of cosine on the same axes as your secant function. c. Write observations of at least three relationships you see between the graphs. 10. What similar relationships will help to graph the cosecant? a. Sketch a graph of y = sin x on the interval -2n s x $ 2n. b. Without using Desmos, use your sine graph to sketch the graph of y = csc x. Start by putting in the asymptotes! 11. How about cotangent? This one's harder to see! c. Start with a graph of y = tan x, from -2n S x S 2n. d. Draw a set of axes below your graph and, again, without Desmos see if you can create the graph of y = cot X. (you can use Desmos to check, but try it first!)