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Transformations of trigonometric functions

Transformations of trigonometric functions. y. y = 2sin x. x. y = sin x. y = sin x. y = – 4 sin x. reflection of y = 4 sin x. y = 4 sin x.

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Transformations of trigonometric functions

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  1. Transformations of trigonometric functions

  2. y y = 2sin x x y = sin x y = sin x y = –4 sin x reflection ofy = 4 sin x y = 4sin x The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| < 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. Amplitude

  3. Thinking about transformations that you learned and knowing whaty = sin xlooks like, what do you supposey = sin x + 2looks like? This is often written with terms traded places so as not to confuse the 2 with part of sine function y = 2 + sin x The function value (or y value) is just moved up 2 units. y = sin x

  4. Thinking about transformations that you've learned and knowing whaty = sin xlooks like, what do you supposey = sin x - 1 looks like? y = sin x The function value (or y value) is just moved down 1. y = - 1 + sin x

  5. Thinking about transformations that you learned and knowing whaty = sin xlooks like, what do you supposey = sin(x + /2) looks like? y = sin x This is a horizontal shift by - /2 y = sin(x + /2)

  6. Amplitude:a Vertical Shift-d Horizontal Shift: c Phase Shift= c/b Equate (bx-c)=0 Phase Shift=c/b

  7. Domain: all real numbers except odd multiples of These are asymptotes. *This is where the cosine is zero. Range: Period: (new period: ) or 180/b Inflection points: Halfway between the vertical asymptotes will be a crossing. Remember this is where the sine is zero. The graph of tangent increases from left to right Important Characteristics of the Tangent Yes…increases from left to right!

  8. y=tan x has the same flipping characteristics as y = sin x. If this graph is flipped it decreases from left to rightTangent has no defined amplitude, since the graph increases (or decreases) without bound “Flipping” (Reflecting) and Amplitude

  9. Domain: all real numbers except multiples of These are asymptotes. *This is where the sine is zero. Range: Period: (new period: ) Inflection points: Halfway between the vertical asymptotes will be a crossing. Remember this is where the cosine is zero. The graph of cotangent decreases from left to right Yes…you heard right, it decreases from left to right! Important Characteristics of the Cotangent

  10. “Flipping” (Reflecting) and Amplitude y=cot x also has the same flipping characteristics as y = sin x. Do NOT flip until the very end. If this graph is flipped it increases from left to right. Cotangent has no defined amplitude, since the graph increases (or decreases) without bound. Flips about X-axis

  11. The graph of secx Here is a sample graph (y = cosx in blue and y = secx in green):

  12. The graph of cosecx .Here is a sample graph (y = sinx is in red and y = cscx is in purple):

  13. http://orion.math.iastate.edu/trig/sp/xm08/applets/tangent.htmlhttp://orion.math.iastate.edu/trig/sp/xm08/applets/tangent.html

  14. Assignment Suggested Questions: Lets Work on Homework Questions 3, 5 and 6 4.6.11

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