1. Transformations of graphs
Trig Functions

2. 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

3. 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)

4. 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

5. 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

6. 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

7. 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

8. 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)