Professor-Leonard
This video focuses on how to graph transformations of the tangent and cotangent functions in precalculus trigonometry. The period for both functions is pi and starts at zero for cotangent but is centered at zero for tangent. The video explains how to identify any vertical shifts and manipulate the key points to graph the transformed function. Key features, such as vertical asymptotes and x-intercepts, are also identified to aid in the graphing process. The video concludes by noting that the next video will focus on the graphs of secant and cosecant functions.
In this section, the instructor discusses how to transform tangent and cotangent functions in terms of a vertical shift, vertical stretch or compression, and altering the period which is a horizontal stretch or compression. The period for both tangent and cotangent is pi, not two pi, and the function's period must be symmetrical. Cotangent is a little more similar to sine and cosine. The first step in transforming these functions is to determine any vertical shift followed by correctly placing the period on the graph, then determining the key points and adjusting the amplitude of the function.
In this section, the instructor teaches how to graph transformations with tangent and cotangent. They first identify if there is any vertical shift and then focus on the period, which for tangent is pi. They divide the period by two to get the interval, and this is centered around zero, which acts as the center of the period for tangent. The next step is to identify key features, such as vertical asymptotes at the ends of the period and x-intercepts at the center. They also find key points at the quarters of the period, where tangent climbs and cotangent falls. This approach enables students to graph the function and identify its key features.
In this section of the video, the instructor explains how to graph transformations using tangent and cotangent functions in precalculus trigonometry. The instructor starts by explaining how to find the key points of the graph, which are affected by the coefficient of the function. Then, students learn how to perform a vertical stretch or compression based on the multiplier of the coefficient and are given an example of how to adjust the graph using pi as a period. The instructor also explains how to find the quarters and center of a different period and places the vertical asymptotes and key points on the graph. The process for graphing tangent and cotangent is similar to the process for graphing sine and cosine.
In this section, the tutor continues the topic of graphing transformations with tangent and cotangent functions, focusing on the vertical shift and the impact of the number by which we divide the period. He emphasizes that the period centers on zero and how to locate key points like the first quarter associated with it. He then uses an example of y equals negative two tangent three x plus one to explain how to apply the transformations. The equation has four transformations - up one, horizontally compressed, vertically stretched, and reflected, and the period is pi over three, centered at zero. Finally, he demonstrates how to find the quarters of the period and locate them on the graph.
In this section, we learn how to graph transformations of tangent functions. First, we identify the key features of tangent at the ends of the period such as the vertical asymptote at the center of the period and the shifted x-intercept. We then deal with the key points by recognizing that tangent has negative one in the first quarter and positive one in the third quarter. We stretch and reflect these points using the negative two factor, which multiplies the outputs. We then base our new key points on the shifted x and y axis and use them to graph the function. If we need to do another cycle, we add the period and then graph the function accordingly.
In this section, the instructor discusses the key points in graphing transformations with tangent and cotangent functions. The first step is to determine if there is a vertical shift, followed by finding the period based on the coefficient of x. The periods for tangent and cotangent are pi, but cotangent starts at zero instead of being centered at zero like tangent. The quarters for tangent and cotangent give us important points that help us manipulate the graph, and for cotangent, the key points are first quarter positive one and third quarter negative one. Lastly, the instructor highlights that the graph of cotangent is downward and reflective, which creates a decreasing graph.
In this section, the video explains how to graph transformations using cotangent in precalculus trigonometry. Cotangent has a period of pi, is not centered at zero but starts at zero, and has vertical asymptotes at the end of the period. The x-intercept is the center of the period, with the first quarter being positive and the second quarter being negative. The video shows how to identify the function, any vertical shifts, and the period, and then how to stretch and manipulate the key points to graph the transformed cotangent function.
In this section, the speaker explains how to graph transformations of the tangent and cotangent functions in precalculus. They identify the period of the function, which is a little larger than the normal period of pi, indicating a slight horizontal stretch. The key features of the graph, such as vertical asymptotes at the start and end and an x-intercept in the center, are identified. The speaker then explains how to find the key points at the quarters, which are adjusted by multiplying, based on the new x-axis created by the shift. They emphasize the importance of identifying the function and its key features to graph its transformations.
In this section, the speaker discusses a technique for graphing transformations using tangent and cotangent functions. The technique includes finding the period, center, and quarters of the graph and then using the features of the graph to plot it. This approach has been effective for many students and is a straightforward process. The video concludes by mentioning that the next video will focus on the graphs of secant and cosecant functions.
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