Amplitude, period and phase shift
Problem 13.95 · medium
For \( \displaystyle y = 2 - 2 \sin{\left(\frac{\pi x}{2} \right)} \), 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 = 2, B = pi/2, h = -1, D = 2.
- \[ \frac{2 \pi}{\frac{1}{2} \pi} = 4 \]Period = 2π/B.✓ Proved
- \[ \left(2 \sin{\left(\frac{\pi x}{2} \right)} - 2\right) + \left. 2 - 2 \sin{\left(\frac{\pi x}{2} \right)} \right|_{\substack{ x=x + 4 }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\left(-1\right) \pi}{2 \frac{\pi}{2}} = -1 \]Phase shift h = C/B.✓ Proved
- \[ \left. 2 - 2 \sin{\left(\frac{\pi x}{2} \right)} \right|_{\substack{ x=-1 }} = 4 \]At x = -1 the curve is 2 from its midline y = 2: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 2,\ \text{period } 4,\ \text{phase shift } 1\text{ to the left},\ \text{vertical shift } 2 \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 setup incorrectly identifies the phase shift as h = –1. Converting 2–2 sin(πx/2) to a cosine form gives 2–2 cos(πx/2–π/2)=2+(-2)cos(π/2(x–1)), so the phase shift is +1 (to the left), not –1. Consequently the subsequent calculations for the phase shift are also wrong.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the phase shift as 1 to the left. The function y = 2 - 2sin(pi*x/2) can be written as 2 + 2sin(pi*x/2 - pi), which corresponds to a phase shift of 2 to the right (or 2 to the left depending on convention, but definitely not 1). The calculation in line 4 computes -1, but this does not correspond to the phase shift of the given sine function relative to a standard sine or cosine without further justification of the identity used. Specifically, -2sin(u) = 2sin(u - pi), so the shift is pi/(pi/2) = 2.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly identifies the phase shift as 1 to the left. The function y = 2 - 2sin(pi*x/2) can be written as 2 + 2sin(pi*x/2 - pi), which corresponds to a phase shift of 2 to the right (or 2 to the left depending on convention, but definitely not 1). The calculation in line 4 computes -1, but this does not correspond to the phase shift of the given sine function relative to a standard sine or cosine without further justification of the identity used. Specifically, -2sin(u) = 2sin(u - pi), so the shift is pi/(pi/2) = 2.gpt-oss:20b: fail (error) 2026-10-17 — The setup incorrectly identifies the phase shift as h = –1. Converting 2–2 sin(πx/2) to a cosine form gives 2–2 cos(πx/2–π/2)=2+(-2)cos(π/2(x–1)), so the phase shift is +1 (to the left), not –1. Consequently the subsequent calculations for the phase shift are also wrong.qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the phase shift as 1 to the left. Converting y = 2 - 2sin(pi*x/2) to cosine form yields y = 2 + 2cos(pi*x/2 + pi/gpt-oss:20b: fail (error) 2026-10-14 — The phase‑shift sentence is incorrect: the function 2-2 sin(πx/2) is equivalent to 2-2 cos(π/2(x-1)), so the shift is 1 to the left, not h = -1 as stated.qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the phase shift as 1 to the left (h = -1). The function y = 2 - 2 sin(pi x / 2) can be written as 2 + 2 sin(pi xgpt-oss:20b: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly rewrites the function as \(2\cos(\frac{\pi}{2}(x-(-1)))+2\). The correct form is \(2-2\sin(\frac{\pi x}{2})=2-2\cos(\frac{\pi}{2}
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 2026-10-17 with SymPy 1.14.0.