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Transformations of graphs

Transformations of graphs. Trig Functions. Features of trig parent graphs. Amplitude. y = a trig x Equation:| a | = amplitude Amplitude is defined for sin and cos only Graph: amplitude = (peak – trough)/2 Transformation: vertical stretch/shrink/reflection

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Transformations of graphs

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  1. Transformationsof graphs Trig Functions

  2. Features of trig parent graphs

  3. Amplitude • y = a trig x • Equation:| a | = amplitude • Amplitude is defined for sin and cos only • Graph: amplitude = (peak – trough)/2 • Transformation: vertical stretch/shrink/reflection • Circle: amplitude equals radius

  4. Period • y = trig (bx) • Normal period for sin, cos, csc and sec is 360˚ or 2π • Normal period for tan and cot is 180˚ or π • Equation: Actual period = Normal period/b Graph: period equals distance between cycles; b = number of cycles between 0˚ and 360˚ for sin,cos,csc or sec; between 0˚ and 180˚ for tan and cot • Transformation: horizontal stretch/shrink/reflection • Circle: b changes rate of rotation (think of gears or pulleys)

  5. Phase shift • y = trig(bx + c) • Equation: Phase shift = -c/b • Graph: horizontal shift (left if c is pos; right if neg) • Circle: Starting at a different point on the circle

  6. Vertical Displacement • y = trig x + d • Equation: Vertical displacement = d • Graph: vertical shift (up if d is pos; down if neg) d = distance that the x-intercepts of the parent graph moved up or down • Circle: The entire circle is moved up or down

  7. Summary of transformations on a rectangular graphy = a trig (bx + c) + d • a • Vertical stretch if |a| > 1 • Vertical shrink if |a| < 1 • Vertical reflection if a < 0 • b • Horizontal shrink if |b| > 1 • Horizontal stretch if |b| < 1 • Horizontal reflection if b < 0 • c • Horizontal shift: left if c > 0, right if c < 0 • d • Vertical shift: up if d > 0, down if d < 0

  8. Summary of trig calculationsy = a trig (bx + c) + d • Amplitude = |a| • Actual period = normal period / |b| • normal period for sin, cos, csc and sec is 360˚ or 2π • normal period for tan and cot is 180˚ or π • Phase shift = -c / b • Vertical displacement = d Finding values from a graph • | a | = amplitude = (peak – trough) / 2 • b = number of cycles between 0 ˚ and 360˚ for sin,cos,csc and sec; between 0 ˚ and 180˚ for tan and cot • c = k * -b where k is the x-value of the point where the y-intercept has moved to • d = distance that the x-intercepts of the parent graph moved up or down

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