Step by Step
D
Vertical shift
D shifts the entire graph up or down. The midline of the graph (the horizontal line the wave oscillates around) is y = D.
A
Amplitude
|A| is the amplitude — how far the graph goes above and below the midline. A larger |A| means a taller wave.
B
Frequency (affects period)
B controls how compressed or stretched the wave is horizontally. The period (length of one full wave cycle) is 2π/B — a larger B means a shorter period (more compressed wave).
C
Phase shift
C shifts the graph left or right. If C is positive, the graph shifts to the right by C units; if negative, it shifts left.
Applied Walkthrough
1
Analyze the equation y = 3 + 2sin(4(x − 1)). Identify all four DABC values: D = 3, A = 2, B = 4, C = 1.
2
The midline is y = 3 (from D), and the amplitude is 2 (from A), meaning the graph oscillates between y = 1 and y = 5.
3
The period is 2π/B = 2π/4 = π/2 — a fairly compressed wave, completing a full cycle every π/2 units along the x-axis.
4
The phase shift is C = 1, meaning the graph is shifted 1 unit to the right compared to a standard sine curve.
Exam Application
Exams test whether you can correctly extract all four DABC values from a sinusoidal equation, and whether you understand how each one individually affects the graph (vertical position, height, horizontal compression, and horizontal position).
⚠ Common Trap
The most common trap is computing the period incorrectly — remember it's 2π/B, not just B itself, and a larger B actually makes the period shorter (a more compressed wave), which can feel counterintuitive.
✓ Quick Self-Check
1. What does the DABC mnemonic represent in y = D + A·sin(B(x−C))?
D = vertical shift, A = amplitude, B = frequency (affects period), C = phase shift.
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2. What is the formula for the period in terms of B?
Period = 2π/B.
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3. For y = 3 + 2sin(4(x−1)), what is the midline?
y = 3.
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4. For y = 3 + 2sin(4(x−1)), what is the amplitude?
2.
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5. For y = 3 + 2sin(4(x−1)), what is the period?
π/2.
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