Amplitude, period and phase shift
Problem 13.470 · medium
For \( \displaystyle y = 1 - 4 \sin{\left(3 x \right)} \), 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 = -4, B = 3, h = pi/6, D = 1.
- Period = 2π/B: 2*pi/3.
- \[ \left(4 \sin{\left(3 x \right)} - 1\right) + \left. 1 - 4 \sin{\left(3 x \right)} \right|_{\substack{ x=x + \frac{2 \pi}{3} }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\pi}{2 \cdot 3} = \frac{\pi}{6} \]Phase shift h = C/B.✓ Proved
- \[ \left. 1 - 4 \sin{\left(3 x \right)} \right|_{\substack{ x=\frac{\pi}{6} }} = -3 \]At x = pi/6 the curve is 4 from its midline y = 1: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } \frac{2 \pi}{3},\ \text{phase shift } \frac{\pi}{6}\text{ to the right},\ \text{vertical shift } 1 \)
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 (error) — The solution incorrectly rewrites the function as a cosine with a phase shift of π/6 and a negative amplitude. The original function y=1-4 sin(3x) has amplitude 4, period 2π/3, no phase shift, and vertical shift 1. The sentences misapply the cosine form and introduce a false phase shift, leading to an incorrect model of the problem.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the phase shift as \pi/6 to the right. The function y = 1 - 4\sin(3x) has a phase shift of 0 (or \pi/3 to the right if converted to positive sine, but certainly not \pi/6). The value \pi/6 is the location of the first minimum, not the phase shift of the standard form. Additionally, the setup in step 1 claims A=-4, B=3, h=\pi/6 for a cosine form, which is inconsistent with the given sine function without proper conversion justification.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution incorrectly identifies the phase shift as \pi/6 to the right. The function y = 1 - 4\sin(3x) has a phase shift of 0 (or \pi/3 to the right if converted to positive sine, but certainly not \pi/6). The value \pi/6 is the location of the first minimum, not the phase shift of the standard form. Additionally, the setup in step 1 claims A=-4, B=3, h=\pi/6 for a cosine form, which is inconsistent with the given sine function without proper conversion justification.gpt-oss:20b: fail (error) 2026-11-08 — The solution incorrectly rewrites the function as a cosine with a phase shift of π/6 and a negative amplitude. The original function y=1-4 sin(3x) has amplitude 4, period 2π/3, no phase shift, and vertical shift 1. The sentences misapply the cosine form and introduce a false phase shift, leading to an incorrect model of the problem.qwen3.6:27b-mlx: fail (error) 2026-11-07 — The solution incorrectly identifies the phase shift as \pi/6 to the right. The function y = 1 - 4\sin(3x) has no horizontal shift relative to the standard sine function (phase shift is 0). The value \pi/6 corresponds to the first positive x-intercept or a specific point on the curve, not the phase shift of the sinusoidal model. Additionally, the setup in step 1 claims A=-4, B=3, h=\pi/6 for a cosine form, which is inconsistent with the derived phase shift and the original sine function.gpt-oss:20b: fail (error) 2026-11-07 — The solution misidentifies the amplitude sign, incorrectly models the function as a cosine with a negative amplitude, and gives the wrong phase‑shift direction. The phase shift should be left by π/6, not right, and the amplitude should be stated as +4. The sentences about the cosine form and the shift calculation do not follow from the given function.qwen3.6:27b-mlx: fail (error) 2026-11-07 — The solution incorrectly identifies the phase shift as pi/6 to the right. The function y = 1 - 4 sin(3x) has a phase shift of 0 (or pi/3 to the left/right depending on convention, but definitely not pi/6 to the right). The step claiming h = pi/6 based on (pi/2)/3 is a confusion between the first zero-crossing/peak location and the phase shift parameter in a standard form, leading to an incorrect final answer.gpt-oss:20b: pass 2026-11-07
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 2026-11-08 with SymPy 1.14.0.