PreCalc 2 · Trigonometric Functions
Look at the start: clouds mean cosine, ground means the sine snake. Then read amplitude, period, and shifts.
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Parent function
y = sin x
sine snake on the ground
y = −sin x
y = cos x
cosine starts in the clouds
y = −cos x
Period 2πB
How far does it take the graph to repeat? This is the period.
Period = 2πB
4π = 2πB
B · 4π = 2π
B = 2π4π
B = 24
B = 12
Amplitude and midline
Midline = 3
Amplitude is the distance (ALWAYS positive) from the midline to the top = 2.
Phase shift
y = sin x normally starts at the midline at (0, 0). Now the start of the graph is at π2.
The graph moved right π2. Remember: for parentheses, opposite of the inside.
y = sin(x − π2)
Write the equation of the graph.
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = sin x
Answer: y = sin x
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = cos x
Answer: y = cos x
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = sin x
Answer: y = sin x
Write the equation of the graph.
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 2
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = 2 sin x
Answer: y = 2 sin x
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 3
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = 3 cos x
Answer: y = 3 cos x
Write the equation of the graph.
The graph starts on the ground, going down. That is −sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
The negative flips the graph.
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = −sin x
Answer: y = −sin x
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 12
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = 12 sin x
Answer: y = 12 sin x
Write the equation of the graph.
The graph starts in the basement. That is −cosine.
y = A cos(B(x − C)) + D
A − amplitude = 4
The negative flips the graph.
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = −4 cos x
Answer: y = −4 cos x
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 12
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = 12 cos x
Answer: y = 12 cos x
Write the equation of the graph.
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 2
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = cos x + 2
Answer: y = cos x + 2
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 4
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = sin x + 4
Answer: y = sin x + 4
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = −3
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = cos x − 3
Answer: y = cos x − 3
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = −1
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = sin x − 1
Answer: y = sin x − 1
Write the equation of the graph.
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
4π = 2πB
B = 2π4π
B = 12
C − phase shift = none
y = sin(12x)
Answer: y = sin(12x)
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
π = 2πB
B = 2ππ
B = 2
C − phase shift = none
y = cos(2x)
Answer: y = cos(2x)
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
π = 2πB
B = 2ππ
B = 2
C − phase shift = none
y = sin(2x)
Answer: y = sin(2x)
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
π2 = 2πB
B = 2ππ2
B = 4
C − phase shift = none
y = cos(4x)
Answer: y = cos(4x)
Write the equation of the graph. The phase shift is the opposite sign of inside the parenthesis. When B is not 1, factor B out before you read C.
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
y = sin(x − π2)
Answer: y = sin(x − π2)
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −3
y = sin(x + 3)
Answer: y = sin(x + 3)
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π
y = cos(x − π)
Answer: y = cos(x − π)
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
y = sin(x + π2)
Answer: y = sin(x + π2)
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
π = 2πB
B = 2ππ
B = 2
A student might write y = sin(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
y = sin(2(x − π2))
Answer: y = sin(2(x − π2))
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
π = 2πB
B = 2ππ
B = 2
A student might write y = cos(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
y = cos(2(x + π2))
Answer: y = cos(2(x + π2))
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
π = 2πB
B = 2ππ
B = 2
A student might write y = sin(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
y = sin(2(x − π4))
Answer: y = sin(2(x − π4))
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Period = 2πB
π = 2πB
B = 2ππ
B = 2
A student might write y = cos(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
y = cos(2(x + π6))
Answer: y = cos(2(x + π6))
Write the equation of the graph. Two of the letters have changed.
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 2
D − midline = 1
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = 2 sin x + 1
Answer: y = 2 sin x + 1
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 2
D − midline = 0
Period = 2πB
π = 2πB
B = 2ππ
B = 2
C − phase shift = none
y = 2 cos(2x)
Answer: y = 2 cos(2x)
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
D − midline = 3
Period = 2πB
π = 2πB
B = 2ππ
B = 2
C − phase shift = none
y = sin(2x) + 3
Answer: y = sin(2x) + 3
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 3
D − midline = 0
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
y = 3 sin(x − π2)
Answer: y = 3 sin(x − π2)
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 1
D − midline = −1
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π
y = cos(x + π) − 1
Answer: y = cos(x + π) − 1
Write the equation of the graph.
The graph starts on the ground, going down. That is −sine.
y = A sin(B(x − C)) + D
A − amplitude = 1
The negative flips the graph.
D − midline = 4
Period = 2πB
2π = 2πB
B = 2π2π
B = 1
C − phase shift = none
y = −sin x + 4
Answer: y = −sin x + 4
Write the equation of the graph. All four letters have changed. Factor out B before you read the phase shift.
Write the equation of the graph.
The graph starts on the ground, going up. That is sine.
y = A sin(B(x − C)) + D
A − amplitude = 2
D − midline = 1
Period = 2πB
π = 2πB
B = 2ππ
B = 2
A student might write y = 2 sin(2x − π) + 1.
Factor out B = 2.
2x − π = 2(x − π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
y = 2 sin(2(x − π2)) + 1
Answer: y = 2 sin(2(x − π2)) + 1
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 3
D − midline = −2
Period = 2πB
π = 2πB
B = 2ππ
B = 2
A student might write y = 3 cos(2x + π) − 2.
Factor out B = 2.
2x + π = 2(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
y = 3 cos(2(x + π2)) − 2
Answer: y = 3 cos(2(x + π2)) − 2
Write the equation of the graph.
The graph starts on the ground, going down. That is −sine.
y = A sin(B(x − C)) + D
A − amplitude = 2
The negative flips the graph.
D − midline = 3
Period = 2πB
π = 2πB
B = 2ππ
B = 2
A student might write y = −2 sin(2x − π2) + 3.
Factor out B = 2.
2x − π2 = 2(x − π4)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π4
y = −2 sin(2(x − π4)) + 3
Answer: y = −2 sin(2(x − π4)) + 3
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 4
D − midline = 1
Period = 2πB
π = 2πB
B = 2ππ
B = 2
A student might write y = 4 cos(2x − π) + 1.
Factor out B = 2.
2x − π = 2(x − π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
y = 4 cos(2(x − π2)) + 1
Answer: y = 4 cos(2(x − π2)) + 1
Write the equation of the graph.
The graph starts on the ground, going down. That is −sine.
y = A sin(B(x − C)) + D
A − amplitude = 3
The negative flips the graph.
D − midline = −1
Period = 2πB
π = 2πB
B = 2ππ
B = 2
A student might write y = −3 sin(2x + π) − 1.
Factor out B = 2.
2x + π = 2(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
y = −3 sin(2(x + π2)) − 1
Answer: y = −3 sin(2(x + π2)) − 1
Write the equation of the graph.
The graph starts in the clouds. That is cosine.
y = A cos(B(x − C)) + D
A − amplitude = 2
D − midline = 5
Period = 2πB
π2 = 2πB
B = 2ππ2
B = 4
A student might write y = 2 cos(4x + 2π) + 5.
Factor out B = 4.
4x + 2π = 4(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
y = 2 cos(4(x + π2)) + 5
Answer: y = 2 cos(4(x + π2)) + 5
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