Trig F07 O’Brien
How to Graph y = a tan(bx − c) + d and y = a cot (bx - c) + d
1.
Identify a, b, c, and d.
2.
5.
Determine if there has been an x-axis reflection: if a is negative, there has been an x-axis reflection.
π
Find the period: period = .
b
1
Find the x-increment: x-increment = ⋅ period .
4
Find the phase shift by setting the argument, bx – c, equal to zero and solving for x:
c
c
c
c
right (if we see bx – c and
is positive) or P.S. =
left (if we see bx + c and
is negative).
P.S. =
b
b
b
b
Find the vertical shift: |d| up if d is positive or |d| down if d is negative.
6.
Identify the left asymptote, the three key points, and the right asymptote.
3a.
b.
4.
cotangent: To find the left asymptote, let x = phase shift, x =
LA: x =
c
b
(x of LA + x-increment, d + a)
c
.
b
(x1 + x-inc, d)
(x2 + x-inc, d – a)
RA: x = x3 + x-inc
Note: The left asymptote of a tangent function is NOT x = phase shift.
π
tangent: To find the left asymptote, set the argument, bx – c, equal to − and solve for x.
2
c π
LA: x = −
(x of LA + x-increment, d – a) (x1 + x-inc, d) (x2 + x-inc, d + a) RA: x = x3 + x-inc
b 2b
7.
Plot the asymptotes and the three key points. Connect the points with a smooth, continuous curve. Extend the graph
to two full periods.
Example 1: y = –3 cot (2x + π) +1
1.
a = –3
b=2
2.
There is an x-axis reflection.
3.
period =
4.
phase shift =
5.
vertical shift = 1 up
6.
LA: x = −
π
2
c = –π
d=1
x-increment =
1 π π
⋅ =
4 2 8
π
left
2
2x + π = 0 → 2x = − π → x = −
π
4π
=−
2
8
⎛ 3π
⎞
, − 2⎟
⎜−
⎝ 8
⎠
⎛ 2π ⎞
, 1⎟
⎜−
⎝ 8
⎠
⎛ π
⎞
⎜ − , 4⎟
⎝ 8
⎠
π
4π
=−
2
8
RA: x = 0
7.
1
Trig F07 O’Brien
⎛1
⎞
Example 2: y = 2 tan ⎜ x − π ⎟ – 3
4
⎝
⎠
b=
1
4
c=π
d = –3
1.
a=2
2.
There is no x-axis reflection.
3.
period =
π
1
= 4π
x-increment =
4
4.
phase shift = 4π right
5.
vertical shift = 3 down
6.
LA:
1
⋅ 4π = π
4
1
x−π =0
4
→
π
1
x−π =−
→ x − 4π = −2π → x = 2 π
4
2
x − 4π = 0 →
(3π, –5)
x = 4π
(4π, –3)
(5π, –1)
RA: x = 6π
7.
2