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Basic Sine and Cosine Functions

Basic Sine and Cosine Functions. Sketch by hand. key points x-intercepts minimums maximums. Sine Function-Keys. Maximum ( /2, 1). Intercept (  , 0). Intercept (2 , 0). Intercept (0, 0). Minimum (3 /2, - 1). Cosine Function --Keys. Maximum (0, 1). Maximum (2  , 1).

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Basic Sine and Cosine Functions

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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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