Transformations of the Sine and Cosine Graphs

Transformations of the Sine and Cosine Graph – An Exploration By Sharon K. O’Kelley This is an exploration for Advanced Algebra or Precalculus teachers who have introduced their students to the basic sine and cosine graphs and now want their students to explore how changes to the equations affect the graphs. This is an introductory lesson whose purpose is to connect the language of Algebraic transformations to the more advanced topic of trignonometry. 1. 2. then the values of a = 1, b = 1, and c = 0. Let’s find out what happens when those values change…. 3. Equation of blue graph Equation of red graph a. b. c. 4. Equation of purple graph Equation of green graph a. b. c. d. 5. b. c. 6. a. b. c. d. 7. a. b. 8. Consider the graph of …. (The first function is in black.) Describe the transformations fully. (Hint: Look at this problem as 9. a. 10. 11. Key to the Exploration 3. a. b. . ) on the red graph. 4. a. b. . c. it is a vertical shrink by . , however, it is a horizontal stretch by a factor of 2. b. Related:  Mathematics

Transformation of Trigonometric Graphs OML Search In these lessons, we will learn how Trigonometric Graphs can be transformed. the amplitude and vertical shift of Trigonometric Graphs the period and phase shift of Trigonometric Graphs Related Topics:More Trigonometric Lessons Stretching and Compressing of Graphs Amplitude of Trigonometric Functions The amplitude of a trigonometric function is the maximum displacement on the graph of that function. In the case of sin and cos functions, this value is the leading coefficient of the function. In the case of tan, cot, sec, and csc, the amplitude would be infinitely large regardless of the value of A. Period of Trigonometric Function The period of a function is the displacement of x at which the graph of the function begins to repeat. Consider y = sin x The value x = 2π is the point at which the graph begins to repeat that of the first quadrant. The general form isy = A sin Bxwhere is the amplitude and B determines the period. Solution: Since B = 2, the period is Solution: Rewrite Videos

Review : Trig Functions The intent of this section is to remind you of some of the more important (from a Calculus standpoint…) topics from a trig class. One of the most important (but not the first) of these topics will be how to use the unit circle. We will actually leave the most important topic to the next section. First let’s start with the six trig functions and how they relate to each other. Recall as well that all the trig functions can be defined in terms of a right triangle. From this right triangle we get the following definitions of the six trig functions. Remembering both the relationship between all six of the trig functions and their right triangle definitions will be useful in this course on occasion. Next, we need to touch on radians. Know this table! Be forewarned, everything in most calculus classes will be done in radians! Let’s next take a look at one of the most overlooked ideas from a trig class. Below is the unit circle with just the first quadrant filled in. and (or , or (start at etc. In fact .

How to Change the Amplitude, Period, and Position of a Tangent or Cotangent Graph You can transform the graph for tangent and cotangent vertically, change the period, shift the graph horizontally, or shift it vertically. However, you should take each transformation one step at a time. For example, to graph follow these steps: Sketch the parent graph for tangent. This graph doesn't shift horizontally, because no constant is added inside the grouping symbols (parentheses) of the function. Now that you've graphed the basics, you can graph a function that has a period change, as in the function You see a lot of pi in that one. Sketch the parent graph for cotangent.Shrink or stretch the parent graph.No constant is multiplying the outside of the function; therefore, you can apply no shrink or stretch.Find the period change.You factor out thewhich affects the period. The transformed graph of y(x) = cot 2pi(x + 1/4).