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3.6 · Standard Form
Write the standard form of a sinusoidal function. What do a, b, and d each represent?
f(θ) = ?
Three transformations encoded in three constants
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3.6 · Solving for b
The period of a sinusoidal function is P. How do you find b?
P = 2π/|b|
Rearrange to isolate b
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3.6 · Example 1
h(θ)=a·sin(bθ)+d has amplitude=6, midline y=3, period=π. Find a, b, and d.
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3.6 · Example 4
k(x) has max at (0,6) and next min at (π/2,−4). Write an expression for k(x).
Half-period=π/2 → period=π · max at x=0 → use cosine
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3.6 · Phase Shift Def
What is a phase shift? How does f(x−c) differ from f(x+c)?
Left or right? Which sign means which?
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3.6 · Example 5
f(x) = 3sin(2x) − 1. State the amplitude, midline, period, max, and min.
Read a, d, and b directly from the equation
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3.6 · Example 7
Graph of h: midline y=1, amplitude=2, period=π, starts at midline going up at x=π/4. Write h(x).
Which trig function? What phase shift?
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3.6 · Example 8 Setup
Five points on h: F(2,16), G(5,11), J(8,6), K(11,11), P(14,16). Find a, b, d.
Max=16, min=6, period=distance between maxima
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✓ Solving for b
b = 2π / P
Rearrange P = 2π/|b|: multiply both sides by b and divide both by P. For P=π: b=2π/π=2. For P=4π: b=2π/(4π)=1/2.
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✓ Standard Form
f(θ) = a·sin(bθ) + d (or a·cos(bθ)+d) · a=amplitude · P=2π/|b| · d=midline
a: vertical dilation (amplitude). b: horizontal dilation (determines period). d: vertical translation (midline). No 'stretch/compress' language on AP exam.
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✓ Example 4 — k(x)
k(x) = 5cos(2x) + 1
Half-period=π/2 → P=π → b=2. Midline=(6+(−4))/2=1 → d=1. Amplitude=6−1=5 → a=5. Max at x=0 → use cosine (cos(0)=1=max). Answer: 5cos(2x)+1.
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✓ Example 1 — Constants
a=6 · b=2 · d=3 → h(θ)=6sin(2θ)+3
a=amplitude=6. d=midline=3. Period=π → b=2π/π=2. If the graph starts at midline going up, use sine.
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✓ Example 5 — Properties
Amplitude=3 · Midline y=−1 · Period=π · Max=2 · Min=−4
a=3, d=−1, b=2→P=π. Max=d+a=−1+3=2. Min=d−a=−1−3=−4. Sine starts at midline going up at x=0.
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✓ Phase Shift
f(x−c): RIGHT by c · f(x+c): LEFT by c
MINUS shifts RIGHT (opposite of what you expect!). Example: sin(x−π/2) is sine shifted right by π/2, which equals cosine. Key: (x−c) → shift right, (x+c) → shift left.
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✓ Example 8 — a, b, d
a=5 · b=π/6 · d=11
Max=16, min=6 → midline=11=d. Amplitude=16−11=5=a. Period=14−2=12 → b=2π/12=π/6.
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✓ Example 7 — Answer
h(x) = 2sin(2(x − π/4)) + 1
Midline=1→d=1. Amplitude=2→a=2. Period=π→b=2. Sine starting at midline going up at x=π/4 → phase shift right by π/4 → (x−π/4).
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3.6 · Solving for c
h(t)=5sin((π/6)(t+c))+11 has its maximum at t=2. Solve for c.
At max: argument of sin = π/2
Set up the equation and solve
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3.6 · sin vs cos
A sinusoidal graph has its maximum at x=0. Should you use sin or cos? Why?
What is sin(0) vs cos(0)?