Amplitude, period and phase shift
Problem 13.268 · medium
For \( \displaystyle y = 4 \cos{\left(\frac{\pi x}{2} \right)} + 3 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.
- Write it as A·sin(B(x − h)) + D: A = 4, B = pi/2, h = -1, D = 3.
- \[ \frac{2 \pi}{\frac{1}{2} \pi} = 4 \]Period = 2π/B.✓ Proved
- \[ \left(- 4 \cos{\left(\frac{\pi x}{2} \right)} - 3\right) + \left. 4 \cos{\left(\frac{\pi x}{2} \right)} + 3 \right|_{\substack{ x=x + 4 }} = 0 \]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
- \[ \left. 4 \cos{\left(\frac{\pi x}{2} \right)} + 3 \right|_{\substack{ x=0 }} = 7 \]At x = 0 the curve is 4 from its midline y = 3: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } 4,\ \text{phase shift } 1\text{ to the left},\ \text{vertical shift } 3 \)
Lines: 4 proved, 1 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 | ✓ 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 solution incorrectly rewrites the cosine function as a sine form and assigns a phase shift of –1. In fact, y=4cos(πx/2)+3 has no horizontal shift (h=0), so the statements about h and the phase‑shift calculation are wrong.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the phase shift as 1 to the left (h = -1). The function y = 4 cos(πx/2) + 3 has no horizontal shift relative to the standard cosine form; the phase shift is 0. The calculation in step 4 appears to be a nonsensical division yielding -1, which does not correspond to any valid parameter extraction for this function.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly identifies the phase shift as 1 to the left (h = -1). The function y = 4 cos(πx/2) + 3 has no horizontal shift relative to the standard cosine form; the phase shift is 0. The calculation in step 4 appears to be a nonsensical division yielding -1, which does not correspond to any valid parameter extraction for this function.gpt-oss:20b: fail (error) 2026-10-18 — The solution incorrectly rewrites the cosine function as a sine form and assigns a phase shift of –1. In fact, y=4cos(πx/2)+3 has no horizontal shift (h=0), so the statements about h and the phase‑shift calculation are wrong.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly identifies the phase shift as 1 to the left (h = -1) when the function y = 4 cos(pi x / 2) + 3 has no phase shift (h = 0) relative to the standard cosine form. The calculation in step 4 is unjustified and does not follow from the given function.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the phase shift as 1 to the left. The function y = 4 cos(pi x / 2) + 3 has no horizontal shift (phase shift is 0) because it is in the standard form A cos(Bx) + D. The solver likely confused the period or misapplied the phase shift formula for a sine conversion without accounting for the phase difference between sine and cosine.gpt-oss:20b: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly rewrites the cosine function as a sine form and assigns a non‑zero phase shift. The correct form is \(4\cos(\tfrac{\pi}{2}x)+3\)
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.