PreCalc 2 · Trigonometric Functions
Cosine starts in the clouds. The sine snake starts on the ground. Then stretch, flip, and shift — one change at a time.
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Cosine starts in the clouds
At x = 0, cosine is at its highest point (the clouds).
Then it comes down, hits the ground, goes to the basement, and climbs back to the clouds.
The sine snake starts on the ground
At x = 0, sine is on the midline (the ground) and slithers up.
Ground → clouds → ground → basement → ground.
Five key points
Period of y = sin x and y = cos x
Period = 2π
There are four parts of a wave, so we divide 2π4.
Each part of the wave is π2 long.
The four letters
y = A sin(B(x − C)) + D or y = A cos(B(x − C)) + D.
|A| = amplitude (how far from the ground to the clouds).
Period = 2π|B| (how long one cycle is).
C = phase shift (slide left or right). D = vertical shift (move the ground).
Example:
y = 2 sin 4(x − 1) + 7
A amplitude = 2
B = 4, so period =
Period = 2πB
Period = 2π4
Period = π2
(One wave is π2 long.)
C phase shift = +1
The phase shift is the opposite sign of inside the parenthesis.
D midline = 7
Sketch one cycle of y = sin x. List the amplitude, midline, period, and phase shift.
Sine snake starts on the ground, going up.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 0, period 2π, phase shift none
Sketch one cycle of y = cos x. List the amplitude, midline, period, and phase shift.
Cosine starts in the clouds.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 0, period 2π, phase shift none
Sketch one cycle of y = sin x. List the amplitude, midline, period, and phase shift.
Sine snake starts on the ground, going up.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 0, period 2π, phase shift none
Sketch one cycle of y = 2 sin x. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 2
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 2, midline y = 0, period 2π, phase shift none
Sketch one cycle of y = 3 cos x. List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 3
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 3, midline y = 0, period 2π, phase shift none
Sketch one cycle of y = −sin x. List the amplitude, midline, period, and phase shift.
The sine snake still starts on the ground, but it slithers down.
y = A sin(B(x − C)) + D
A − amplitude = 1
The negative flips the graph.
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 0, period 2π, phase shift none
Sketch one cycle of y = 12 sin x. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 12
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 12, midline y = 0, period 2π, phase shift none
Sketch one cycle of y = −4 cos x. List the amplitude, midline, period, and phase shift.
Flipped cosine starts in the basement.
y = A cos(B(x − C)) + D
A − amplitude = 4
The negative flips the graph.
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 4, midline y = 0, period 2π, phase shift none
Sketch one cycle of y = 12 cos x. List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 12
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 12, midline y = 0, period 2π, phase shift none
Sketch one cycle of y = cos x + 2. List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 2
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 2, period 2π, phase shift none
Sketch one cycle of y = sin x + 4. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 4
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 4, period 2π, phase shift none
Sketch one cycle of y = cos x − 3. List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = −3
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = −3, period 2π, phase shift none
Sketch one cycle of y = sin x − 1. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = −1
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = −1, period 2π, phase shift none
Sketch one cycle of y = sin(2x). List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 1, midline y = 0, period π, phase shift none
Sketch one cycle of y = cos(2x). List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 1, midline y = 0, period π, phase shift none
Sketch one cycle of y = sin(12x). List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 12
Period = 2π12
Period = 4π
To graph, divide the period by 4
4π4 = π
The graph changes every π.
Answer: amplitude 1, midline y = 0, period 4π, phase shift none
Sketch one cycle of y = cos(4x). List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 4
Period = 2π4
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Answer: amplitude 1, midline y = 0, period π2, phase shift none
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 = sin(x − π2). List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 0, period 2π, phase shift π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = sin(x + 3). List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −3
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 0, period 2π, phase shift −3
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = cos(x − π). List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 0, period 2π, phase shift π
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = sin(x + π2). List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 0, period 2π, phase shift −π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = sin(2x − π). List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 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 = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 1, midline y = 0, period π, phase shift π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = cos(2x + π). List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 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 = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 1, midline y = 0, period π, phase shift −π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = sin(2x − π2). List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 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 = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 1, midline y = 0, period π, phase shift π4
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = cos(2x + π3). List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 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 = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 1, midline y = 0, period π, phase shift −π6
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Two of the letters have changed.
Sketch one cycle of y = 2 sin x + 1. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 2
D − midline = 1
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 2, midline y = 1, period 2π, phase shift none
Sketch one cycle of y = 2 cos(2x). List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 2
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 2, midline y = 0, period π, phase shift none
Sketch one cycle of y = sin(2x) + 3. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 3
C − phase shift = none
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 1, midline y = 3, period π, phase shift none
Sketch one cycle of y = 3 sin(x − π2). List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 3
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 3, midline y = 0, period 2π, phase shift π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = cos(x + π) − 1. List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = −1
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = −1, period 2π, phase shift −π
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = −sin x + 4. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 1
The negative flips the graph.
D − midline = 4
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Answer: amplitude 1, midline y = 4, period 2π, phase shift none
All four letters have changed. Factor out B before you read the phase shift.
Sketch one cycle of y = 2 sin(2x − π) + 1. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 2
D − midline = 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 = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 2, midline y = 1, period π, phase shift π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = 3 cos(2x + π) − 2. List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 3
D − midline = −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 = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 3, midline y = −2, period π, phase shift −π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = −2 sin(2x − π2) + 3. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 2
The negative flips the graph.
D − midline = 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 = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 2, midline y = 3, period π, phase shift π4
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = 4 cos(2x − π) + 1. List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 4
D − midline = 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 = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 4, midline y = 1, period π, phase shift π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = −3 sin(2x + π) − 1. List the amplitude, midline, period, and phase shift.
y = A sin(B(x − C)) + D
A − amplitude = 3
The negative flips the graph.
D − midline = −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 = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Answer: amplitude 3, midline y = −1, period π, phase shift −π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
Sketch one cycle of y = 2 cos(4x + 2π) + 5. List the amplitude, midline, period, and phase shift.
y = A cos(B(x − C)) + D
A − amplitude = 2
D − midline = 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 = 2π|B|
B = 4
Period = 2π4
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Answer: amplitude 2, midline y = 5, period π2, phase shift −π2
Dotted wave: before the phase shift. Solid wave: after the phase shift.
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