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Graphs of Sine and Cosine Functions

Graphs of Sine and Cosine Functions. Section 4.5. Fast Wave, Slow Wave. Who cares about waves??. Create the sine and cosine Graphs. Creating the Graphs. sine and cosine graphs. Sine graph plotted Cosine graph plotted. What about negative values?. Does the graph repeat? When?.

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Graphs of Sine and Cosine Functions

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  1. Graphs of Sine and Cosine Functions Section 4.5

  2. Fast Wave, Slow Wave Who cares about waves??

  3. Create the sine and cosine Graphs Creating the Graphs

  4. sine and cosine graphs Sine graph plotted Cosine graph plotted

  5. What about negative values?

  6. Does the graph repeat?When? One complete cycle of the cosine and sine graphs is called the ___period___

  7. Key Points What did you find??? Key Points What are key points? What do they mean?

  8. Review – Graphing sine and cosine • Where do they graphs come from?!? • Graphing Sine and Cosine – Review • graphing sine and cosine - radians

  9. Listing the Key Points Sinx:(0°,0) (90°, 1) (180°,0) (270°, -1) (360°, 0) Cosx:(0°,1) (90°,0) (180°, -1) (270°, 0) (360°, -1)

  10. Transformations to the parent graphs – sine and cosine • We will see how transformations affect the basic sine and cosine parent graphs

  11. General Equations: • y = asin(bx – c) + d • y = acos(bx – c) + d

  12. Vertical Translations • A vertical shift is the vertical distance between the midline of the graph and the x –axis. • For y = sinx +d and y = cosx +d, the constant d causes a vertical shift in the graph • Which value (x or y) is influenced by the change in the “d” value?

  13. What’s Happening?? Transformations of Sine and Cosine

  14. Vertical Translations • The shift is d units upward for d >0 • The shift is d units downward for d< 0 • The graph oscillates about line y = d instead of x-axis • Does the period change when “d” changes?

  15. Examples:

  16. Amplitude • The amplitude of y = asinx and y = acosx represents the vertical distance between the midline and the maximum or minimum • Amplitude = |a| • The constant factor “a” is a scaling factor - a vertical stretch or shrink of the basic sine and cosine curves

  17. What’s Happening?? Transformations of Sine and Cosine

  18. Amplitude…What Does It Do?? • If a ≥ 1, the basic sine curve is stretched vertically • If a ≤ -1, the basic sine curve is reflected across the x-axis and vertically stretched • The graph of y = asinx ranges between -a and a instead of between -1 and 1

  19. Amplitude…What Does It Do?? • If 0 <a < 1, the basic sine curve is vertically shrunk • If -1< a < 0, the basic sine curve is reflected across the x-axis and vertically shrunk • The graph of y = asinx ranges between -a and a instead of between -1 and 1

  20. Examples:

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