Transformations of graphs Trig Functions

Amplitude y = a·trig(bx + c) + d Equation: amplitude = | a | Amplitude is defined for sin and cos only Transformation: vertical stretch/shrink/reflection Graph: amplitude = (max – min)/2 Circle: amplitude equals radius

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

Phase shift y = a·trig(bx + c) + d Equation: Phase shift = -c/b Transformation: horizontal shift (left if c is pos; right if neg) Graph: Distance the y-intercept moved right or left? Circle: Starting at a different point on the circle

Vertical Displacement y = a·trig(bx + c) + d Equation: Vertical displacement = d Transformation: vertical shift (up if d is pos; down if neg) Graph: d = (max + min) / 2 or distance the x-intercepts moved up or down Circle: The entire circle is moved up or down

Summary of transformations on a rectangular graph y = a trig (bx + c) + d 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

Summary of trig calculations y = a trig (bx + c) + d Amplitude = |a|(amplitude defined for sin & cos only) Actual period = normal period / |b| normal period is 360° (2π) for sin, cos, csc and sec normal period is 180° (π) for tan and cot Phase shift = -c / b Vertical displacement = d Finding values from a graph | a | = amplitude = (max – min)/2 (amplitude: sin & cos only) b = number of cycles between 0 and 360° (2π) for sin,cos,csc & sec; between 0 and 180° (π) for tan and cot c = k * -b where k is the distance the y-intercept moved right or left d = (max + min) / 2 -or- the distance the x-intercepts moved up or down

Features of trig parent graphs y=sin(x) First period 0≤ x≤2π 0º≤ x≤360º Domain All real #s Range -1≤ y≤1 y=arcsin(x) -1≤ x≤1 -π/2≤ y≤π/2 -90º≤ y≤90º y=cos(x) y=arccos(x) 0≤ y≤π 0º≤ y≤180º y=csc(x) x≠πk x≠90º(2k) y≤-1 or y ≥1 y=arccsc(x) x≤-1 or x ≥1 y ≠ 0 or 0º y=sec(x) -π/2≤ x≤3π/2 -90º≤ x≤270º x≠π/2(2k+1) x≠90º (2k +1) 1 y=arcsec(x) y ≠ π/2 or 90º y=tan(x) -π/2≤ x≤π/2 -90º≤ x≤90º y=arctan(x) y=cot(x) 0≤ x≤π 0º≤ x≤180º x≠π/2(2k) y=arccot(x)