PreCalc 2 · Trigonometric Functions
Period is π. Mark the dashed asymptotes first, then draw the branch that goes through the middle of the period.
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Tangent
Period is π (half of sine’s period).
It passes through the origin and goes up through Quadrant I.
Dashed lines are vertical asymptotes, at odd multiples of π2 for y = tan x.
Cotangent
Period is also π.
Asymptotes at multiples of π. It falls through (π2, 0).
Four key parts
Period of y = tan x and y = cot x
Period = π
There are four parts of a cycle, so we divide π4.
Each part is π4 long.
The four letters
y = A tan(B(x − C)) + D or y = A cot(B(x − C)) + D.
A = stretch (tan and cot have no highest or lowest point).
Period = π|B| (how long one cycle is).
C = phase shift (slide left or right). D = vertical shift.
Example:
y = 2 tan 4(x − 1) + 7
A stretch = 2
B = 4, so period =
Period = πB
Period = π4
Period = π4
(One cycle is π4 long.)
C phase shift = +1
The phase shift is the opposite sign of inside the parenthesis.
D vertical shift = 7
Sketch one cycle of y = tan x. List the stretch, vertical shift, period, phase shift, and asymptotes.
Tangent goes through (0, 0) and rises.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π
Answer: stretch 1, vertical shift y = 0, period π, phase shift none, asymptotes x = π2 + k·π
Sketch one cycle of y = cot x. List the stretch, vertical shift, period, phase shift, and asymptotes.
Cotangent has asymptotes at 0 and π, and falls through the middle.
y = A cot(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = k·π
Answer: stretch 1, vertical shift y = 0, period π, phase shift none, asymptotes x = k·π
Sketch one cycle of y = tan x. List the stretch, vertical shift, period, phase shift, and asymptotes.
Tangent goes through (0, 0) and rises.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π
Answer: stretch 1, vertical shift y = 0, period π, phase shift none, asymptotes x = π2 + k·π
Sketch one cycle of y = 2 tan x. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 2
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π
Answer: stretch 2, vertical shift y = 0, period π, phase shift none, asymptotes x = π2 + k·π
Sketch one cycle of y = 3 cot x. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 3
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = k·π
Answer: stretch 3, vertical shift y = 0, period π, phase shift none, asymptotes x = k·π
Sketch one cycle of y = −tan x. List the stretch, vertical shift, period, phase shift, and asymptotes.
The negative flips the graph. It still goes through (0, 0), but it falls.
y = A tan(B(x − C)) + D
A − stretch = 1
The negative flips the graph.
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π
Answer: stretch 1, vertical shift y = 0, period π, phase shift none, asymptotes x = π2 + k·π
Sketch one cycle of y = 12 tan x. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 12
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π
Answer: stretch 12, vertical shift y = 0, period π, phase shift none, asymptotes x = π2 + k·π
Sketch one cycle of y = −4 cot x. List the stretch, vertical shift, period, phase shift, and asymptotes.
The negative flips cotangent: it rises through the middle.
y = A cot(B(x − C)) + D
A − stretch = 4
The negative flips the graph.
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = k·π
Answer: stretch 4, vertical shift y = 0, period π, phase shift none, asymptotes x = k·π
Sketch one cycle of y = 12 cot x. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 12
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = k·π
Answer: stretch 12, vertical shift y = 0, period π, phase shift none, asymptotes x = k·π
Sketch one cycle of y = tan x + 2. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 2
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π
Answer: stretch 1, vertical shift y = 2, period π, phase shift none, asymptotes x = π2 + k·π
Sketch one cycle of y = cot x + 4. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 1
D − vertical shift = 4
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = k·π
Answer: stretch 1, vertical shift y = 4, period π, phase shift none, asymptotes x = k·π
Sketch one cycle of y = tan x − 3. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = −3
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π
Answer: stretch 1, vertical shift y = −3, period π, phase shift none, asymptotes x = π2 + k·π
Sketch one cycle of y = cot x − 1. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 1
D − vertical shift = −1
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = k·π
Answer: stretch 1, vertical shift y = −1, period π, phase shift none, asymptotes x = k·π
Sketch one cycle of y = tan(2x). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = π4 + k·π2
Answer: stretch 1, vertical shift y = 0, period π2, phase shift none, asymptotes x = π4 + k·π2
Sketch one cycle of y = cot(2x). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = k·π2
Answer: stretch 1, vertical shift y = 0, period π2, phase shift none, asymptotes x = k·π2
Sketch one cycle of y = tan(12x). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 12
Period = π12
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π + k·2π
Answer: stretch 1, vertical shift y = 0, period 2π, phase shift none, asymptotes x = π + k·2π
Sketch one cycle of y = cot(4x). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 4
Period = π4
Period = π4
To graph, divide the period by 4
π44 = π16
The graph changes every π16.
Asymptotes: x = k·π4
Answer: stretch 1, vertical shift y = 0, period π4, phase shift none, asymptotes x = k·π4
The phase shift is the opposite sign of inside the parenthesis. A shift left 3 is −3. When B is not 1, factor B out before you read C.
Sketch one cycle of y = tan(x − π4). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π4
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = 3π4 + k·π
Answer: stretch 1, vertical shift y = 0, period π, phase shift π4, asymptotes x = 3π4 + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = tan(x + 3). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −3
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = −1.4292 + k·π
Answer: stretch 1, vertical shift y = 0, period π, phase shift −3, asymptotes x = −1.4292 + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = cot(x − π). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π + k·π
Answer: stretch 1, vertical shift y = 0, period π, phase shift π, asymptotes x = π + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = tan(x + π4). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π4
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π4 + k·π
Answer: stretch 1, vertical shift y = 0, period π, phase shift −π4, asymptotes x = π4 + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = tan(2x − π). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
Inside: 2x − π
Factor out B = 2.
2x − π = 2(x − π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = 3π4 + k·π2
Answer: stretch 1, vertical shift y = 0, period π2, phase shift π2, asymptotes x = 3π4 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = cot(2x + π). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
Inside: 2x + π
Factor out B = 2.
2x + π = 2(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = −π2 + k·π2
Answer: stretch 1, vertical shift y = 0, period π2, phase shift −π2, asymptotes x = −π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = tan(2x − π2). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
Inside: 2x − π2
Factor out B = 2.
2x − π2 = 2(x − π4)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π4
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = π2 + k·π2
Answer: stretch 1, vertical shift y = 0, period π2, phase shift π4, asymptotes x = π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = cot(2x + π3). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 1
D − vertical shift = 0
Inside: 2x + π3
Factor out B = 2.
2x + π3 = 2(x + π6)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π6
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = −π6 + k·π2
Answer: stretch 1, vertical shift y = 0, period π2, phase shift −π6, asymptotes x = −π6 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Two of the letters have changed.
Sketch one cycle of y = 2 tan x + 1. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 2
D − vertical shift = 1
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π
Answer: stretch 2, vertical shift y = 1, period π, phase shift none, asymptotes x = π2 + k·π
Sketch one cycle of y = 2 cot(2x). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 2
D − vertical shift = 0
C − phase shift = none
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = k·π2
Answer: stretch 2, vertical shift y = 0, period π2, phase shift none, asymptotes x = k·π2
Sketch one cycle of y = tan(2x) + 3. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
D − vertical shift = 3
C − phase shift = none
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = π4 + k·π2
Answer: stretch 1, vertical shift y = 3, period π2, phase shift none, asymptotes x = π4 + k·π2
Sketch one cycle of y = 3 tan(x − π4). List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 3
D − vertical shift = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π4
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = 3π4 + k·π
Answer: stretch 3, vertical shift y = 0, period π, phase shift π4, asymptotes x = 3π4 + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = cot(x + π) − 1. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 1
D − vertical shift = −1
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = −π + k·π
Answer: stretch 1, vertical shift y = −1, period π, phase shift −π, asymptotes x = −π + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = −tan x + 4. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 1
The negative flips the graph.
D − vertical shift = 4
C − phase shift = none
Period = π|B|
B = 1
Period = π1
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π
Answer: stretch 1, vertical shift y = 4, period π, phase shift none, asymptotes x = π2 + k·π
All four letters have changed. Factor out B before you read the phase shift.
Sketch one cycle of y = 2 tan(2x − π) + 1. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 2
D − vertical shift = 1
Inside: 2x − π
Factor out B = 2.
2x − π = 2(x − π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = 3π4 + k·π2
Answer: stretch 2, vertical shift y = 1, period π2, phase shift π2, asymptotes x = 3π4 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = 3 cot(2x + π) − 2. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 3
D − vertical shift = −2
Inside: 2x + π
Factor out B = 2.
2x + π = 2(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = −π2 + k·π2
Answer: stretch 3, vertical shift y = −2, period π2, phase shift −π2, asymptotes x = −π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = −2 tan(2x − π2) + 3. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 2
The negative flips the graph.
D − vertical shift = 3
Inside: 2x − π2
Factor out B = 2.
2x − π2 = 2(x − π4)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π4
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = π2 + k·π2
Answer: stretch 2, vertical shift y = 3, period π2, phase shift π4, asymptotes x = π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = 4 cot(2x − π) + 1. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 4
D − vertical shift = 1
Inside: 2x − π
Factor out B = 2.
2x − π = 2(x − π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = π2 + k·π2
Answer: stretch 4, vertical shift y = 1, period π2, phase shift π2, asymptotes x = π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = −3 tan(2x + π) − 1. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A tan(B(x − C)) + D
A − stretch = 3
The negative flips the graph.
D − vertical shift = −1
Inside: 2x + π
Factor out B = 2.
2x + π = 2(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = π|B|
B = 2
Period = π2
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = −π4 + k·π2
Answer: stretch 3, vertical shift y = −1, period π2, phase shift −π2, asymptotes x = −π4 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = 2 cot(4x + 2π) + 5. List the stretch, vertical shift, period, phase shift, and asymptotes.
y = A cot(B(x − C)) + D
A − stretch = 2
D − vertical shift = 5
Inside: 4x + 2π
Factor out B = 4.
4x + 2π = 4(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = π|B|
B = 4
Period = π4
Period = π4
To graph, divide the period by 4
π44 = π16
The graph changes every π16.
Asymptotes: x = −π2 + k·π4
Answer: stretch 2, vertical shift y = 5, period π4, phase shift −π2, asymptotes x = −π2 + k·π4
Dotted graph: before the phase shift. Solid graph: after the phase shift.
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