Amplitude, period and phase shift
Problem 13.267 · medium
For \( \displaystyle y = - \sin{\left(3 x \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.
- Write it as A·cos(B(x − h)) + D: A = -1, B = 3, h = pi/6, D = 0.
- Period = 2π/B: 2*pi/3.
- \[ \sin{\left(3 x \right)} + \left. - \sin{\left(3 x \right)} \right|_{\substack{ x=x + \frac{2 \pi}{3} }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\pi}{2 \cdot 3} = \frac{\pi}{6} \]Phase shift h = C/B.✓ Proved
- \[ \left. - \sin{\left(3 x \right)} \right|_{\substack{ x=\frac{\pi}{6} }} = -1 \]At x = pi/6 the curve is 1 from its midline y = 0: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 1,\ \text{period } \frac{2 \pi}{3},\ \text{phase shift } \frac{\pi}{6}\text{ to the right},\ \text{vertical shift } 0 \)
Lines: 3 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | max and min over 16000 samples give the amplitude and midline; f(x + p) = f(x) for the period and not half of it |
Reviewers
gpt-oss:20b: fail (style) — [domain objection, downgraded to style] The solution misidentifies the amplitude and phase shift. It sets A = –1, which is incorrect because amplitude must be non‑negative; the correct amplitude is 1. The phase shift is rightward by π/6, but the derivation uses an undefined C and incorrectly evaluates the function at x = π/6, yielding –1 (the minimum) instead of demonstrating the amplitude. The cosine form is also inappropriate for the given sine function without proper conversion.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the phase shift as pi/6 to the right. For y = -sin(3x), which is equivalent to sin(3x + pi), the phase shift is -pi/3 (pi/3 to the left). The value pi/6 corresponds to a specific point on the graph (a minimum), not the phase shift of the standard form.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly identifies the phase shift as pi/6 to the right. For y = -sin(3x), which is equivalent to sin(3x + pi), the phase shift is -pi/3 (pi/3 to the left). The value pi/6 corresponds to a specific point on the graph (a minimum), not the phase shift of the standard form.gpt-oss:20b: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution misidentifies the amplitude and phase shift. It sets A = –1, which is incorrect because amplitude must be non‑negative; the correct amplitude is 1. The phase shift is rightward by π/6, but the derivation uses an undefined C and incorrectly evaluates the function at x = π/6, yielding –1 (the minimum) instead of demonstrating the amplitude. The cosine form is also inappropriate for the given sine function without proper conversion.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly identifies the phase shift as pi/6 to the right. For y = -sin(3x), the standard form y = A sin(B(x - h)) yields h = 0 (since -sin(3x) = sin(3x + pi) = sin(3(x + pi/3)), implying a shift of -pi/3, or 0 if considering the negative amplitude). Even if converting to cosine, -sin(3x) = cos(3x + pi/2) = cos(3(x + pi/6)), which is a shift of pi/6 to the LEFT, not right. The claim that the shift is to the right is factually wrong.gpt-oss:20b: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly rewrites \(-\sin(3x)\) as \(A\cos(B(x-h))+D\) with \(A=-1\), \(B=3\), \(h=\pi/6\), \(D=0\). This form is not equivalent to the giqwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the phase shift as \pi/6 to the right. The function y = -\sin(3x) can be written as \sin(3x + \pi), which corresponds to a phase shift of -\pi/3 (\pi/3 to the left). The value \pi/6 is the location of the first minimum, not the phase shift of the standard sine wave form.gpt-oss:20b: fail (error) 2026-10-15 — The solution incorrectly rewrites the function as A·cos(B(x−h))+D with A = −1, B = 3, h = π/6, D = 0. The given function is y = −sin(3x), which has amplitude |A| = 1, period 2π/3, no phase shift (h = 0), and vertical shift 0. Thus the setup and the claimed phase shift are wrong.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/sinusoid_features, checked 2026-10-18 with SymPy 1.14.0.