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Understanding Sine and Cosine Functions: Key Points and Graphing Techniques

This guide explores sine and cosine functions, focusing on key points, intercepts, maximums, and minimums. It details the analytical methods for determining the amplitude and period of these functions and discusses how to sketch their graphs by hand. Techniques for translating these functions horizontally and vertically are also covered, demonstrating how changes in parameters affect their graphs. The content includes examples of various transformations, periods, and visual representations, making it a comprehensive resource for mastering sine and cosine functions.

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Understanding Sine and Cosine Functions: Key Points and Graphing Techniques

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  1. Basic Sine and Cosine Functions

  2. Sketch by hand • key points • x-intercepts • minimums • maximums

  3. Sine Function-Keys Maximum (/2, 1) Intercept ( , 0) Intercept (2, 0) Intercept (0, 0) Minimum (3/2, - 1)

  4. Cosine Function --Keys Maximum (0, 1) Maximum (2, 1) Intercept (3/2, 0) Intercept (/2, 0) Minimum (, -1)

  5. Sketch the graph of y = 2 sin x • first remember y = 2 sin x means • y = 2(sin x)

  6. Amplitude • y = a sin x • y = a cos x • a is a scaling factor or amplitude • |a| > 1 is a vertical stretch • |a| < 1 is a vertical shrink • range is –a ≤ y ≤ a

  7. Period • is the distance from any point on the graph to a corresponding point on the graph • usually measured from a key point • period = 2/b • y = a sin bx • y = a cos bx • b = 1 on parent functions

  8. Period • if 0 < b < 1, the period is greater than 2 • and represents a horizontal stretch • if b > 1, the period is less than 2 • and represents a horizontal shrink

  9. Find the period • y = sin x/2 • period = 2/b • x/2 = (1/2)x = bx • so b = ½ so • period = 2/(1/2) = 4 • y = cos x/4 • period = 8 • y = sin 2x • Period = 

  10. Horizontal stretching • sketch y = sin x/2 • determine period • plot x = period • divide x-axis into 4 even parts • sketch the parent function over the new period

  11. Translations • y = a sin (bx – c) and y = a cos (bx – c) • creates a horizontal translation of the parent functions of sine and cosine • the graph completes one cycle from • bx – c = 0, the left end point • to bx – c = 2, the right end point

  12. Find the left and right endpoint, also the period • y = sin(x - /3) • y = cos(2x - /4) • y = cos(x/2 - /4)

  13. Sketch y = sin(x - /3)

  14. Graph y = cos(2x - /4)

  15. Graph cos(x/2 - /4)

  16. What is the difference between • y = sin(x/3 + ) • y = 2 + sin(x/3 + ) • since the 2 is being added to the y value • the entire graph is translated up 2 (y) units • y = cos (2x + /2) • y = - 3 + cos (2x + /2) • the entire graph is translated down 3 units

  17. y = d + a sin (bx – c) • d is the vertical translation • a is the amplitude, 2a is the total vertical movement • b modifies the period, period = 2/b • c is the horizontal translation • bx – c = 0 is the left endpoint of a period • bx – c = 2 is the right endpoint of a period • all of the above is true for cosine

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