General Principles of Intellectual Property: Concepts of Intellectual Proper...
Graphs of trigonometric functions
1. TARUNGEHLOT
Graphs of Trigonometric Functions
Sine Cosine
Period = 2 Period = 2
y = a sin (bx + c) y = a cos (bx + c)
amplitude = a amplitude = a
2 2
period = period =
b b
c c
phase shift = phase shift =
b b
one cycle can be found by solving: one cycle can be found by solving:
0 bx c 2 0 bx c 2
Tangent Cotangent
Period = Period =
x –intercepts at n
x –intercepts at n
2
vertical asymptotes at x = n
2 vertical asymptotes at x = n
y = a tan (bx + c) y = a cot (bx + c)
2.
period = period =
b b
c c
phase shift = phase shift =
b b
successive vert. asymptotes for one branch: successive vert. asymptotes for one branch:
2 bx c 2 0 bx c
Cosecant Secant
Period = 2 Period = 2
Vertical asymptotes at x = n
Vertical asymptotes at x = n
2
y = a csc (bx + c) y = a sec (bx + c)
2 2
period = period =
b b
c c
phase shift = phase shift =
b b
one cycle can be found by solving: one cycle can be found by solving:
0 bx c 2 3
bx c
2 2
To graph y = a csc (bx + c):
First graph y = a sin (bx + c); draw the To graph y = a sec (bx + c):
vertical asymptotes at the x-intercepts; vertical asymptotes at the x-ints,
First graph y = a cos (bx + c); draw the
take the reciprocals. vertical asymptotes at the x-intercepts; vertical as
take the reciprocals.
Summary:
period x-intercepts y-intercepts Vertical asymptotes
y = sin x 2 n 0 none
y = cos x 2 1 none
n
2
y = tan x n 0
x n
2
3. y = cot x none x n
n
2
y = sec x 2 none 1
x n
2
y = csc x 2 none none x n