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. (A key follows the end of the exploration.) 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 . is equivalent to
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 .
Radians to degrees
How to convert degrees to radians or radians to degrees. Theory: What are 'radians' ? One radian is the angle of an arc created by wrapping the radius of a circle around its circumference. In this diagram, the radius has been wrapped around the circumference to create an angle of 1 radian. The pink lines show the radius being moved from the inside of the circle to the outside: The radius 'r' fits around the circumference of a circle exactly 2p times. circumference = 2pr So there are 2p radians in a complete circle, and p radians in a half circle. Converting radians to degrees: To convert radians to degrees, we make use of the fact that p radians equals one half circle, or 180º. This means that if we divide radians by p, the answer is the number of half circles. So, to convert radians to degrees, multiply by 180/p, like this: Converting degrees to radians: To convert degrees to radians, first find the number of half circles in the answer by dividing by 180º. Method:
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).
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Scaffolded Math and Science: Middle School Math Word Wall Ideas
For a year between teaching mainstream high school math and teaching special education high school math, I tried teaching middle school math. My husband and I had moved further out from the city, the job was listed as an 8th grade Algebra position, I love Algebra, it was closer to my new apartment, what could go wrong?Everything. Everything could go wrong. 8th Grade Math Word Wall I made this 8th grade math word wall with thoughts of my former 8th graders in mind. Scatter plots even come up in our 10th grade state exit exam here in Massachusetts. Here is a reference section for function vs. not a function. And transformations. 7th Grade Math Word Wall There are way more representations of slope that I had ever imagined before making this word wall for 7th grade! Here is a reference for scale factor. And integers. And Geometry! 6th Grade Math Word Wall Data, data, data! Here is a box and whiskers reference. Multiplying and dividing fractions. And nets. 5th Grade Math Word Wall Adding fractions.
TESOL Event Detail - TESOL Core Certificate Program (Cohort 22) - More About the Program
Set yourself apart and teach almost anywhere in the world with a certificate from TESOL International Association, the globally-recognized trusted provider of ELT professional learning. The TESOL Core Certificate Program (TCCP) is aligned to TESOL’s Standards for Short-Term TEFL/TESL Certificate Programs and was developed by experts in the TESOL field. Gain foundational knowledge in the theory and practice of English language teaching (ELT), focus your skills on teaching adult or young learners, and apply what you’ve learned through an in-person or virtual teaching practicum. The TESOL Core Certificate Program (TCCP) is a 140-hour blended-learning program, meaning that a portion of the program is completed online and a portion of the program is completed in person. Cohort 22, Starts September 2019 Foundations Course: 4 September – 15 October 2019Specialty Course (Adult or Young Learners): 30 October – 10 December 2019Teaching Practicum: to be completed by 17 July 2020
Understanding Language
This eighth grade (middle school) task provides students opportunities to interpret a situation, represent the variables mathematically, select appropriate mathematical methods, interpret and evaluate the data generated, and communicate their reasoning. Students work with selected formulas to model a situation, interpret given data, make approximations, communicate their reasoning in verbal and written form, and critique solutions developed by others. This task draws on understandings of rate and proportional reasoning (a CCSS focus of grades 6 and 7), geometric measurement and volume (which begins with right rectangular prisms in grade 5, and is extended in grades 6, 7, and 8), and builds toward the high school number and quantity standard of interpreting units consistently in formulas. Making Matchsticks is part of a Formative Assessment Lesson (FAL), which can be downloaded here:
