Amplitude, period and phase shift
Problem 13.273 · medium
For \( \displaystyle y = 4 \cos{\left(2 x - 2 \right)} \), 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 = 4, B = 2, h = 1, D = 0.
- \[ \frac{2 \pi}{2} = \pi \]Period = 2π/B.✓ Proved
- \[ - 4 \cos{\left(2 x - 2 \right)} + \left. 4 \cos{\left(2 x - 2 \right)} \right|_{\substack{ x=x + \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{2}{2} = 1 \]Phase shift h = C/B.✓ Proved
- \[ \left. 4 \cos{\left(2 x - 2 \right)} \right|_{\substack{ x=1 }} = 4 \]At x = 1 the curve is 4 from its midline y = 0: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } \pi,\ \text{phase shift } 1\text{ to the right},\ \text{vertical shift } 0 \)
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 (misleading) — The solution incorrectly derives the phase shift by using 2/2=1 and misapplies the shift test; it also presents a confusing and mathematically incorrect equation for the period shift. The explanation for the amplitude is correct, but the overall reasoning is misleading.qwen3.6:27b-mlx: pass — The solution correctly identifies the parameters A, B, h, and D by rewriting the function in standard form. The derived values for amplitude, period, phase shift, and vertical shift are correct and consistent with the problem statement.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the parameters A, B, h, and D by rewriting the function in standard form. The derived values for amplitude, period, phase shift, and vertical shift are correct and consistent with the problem statement.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution incorrectly derives the phase shift by using 2/2=1 and misapplies the shift test; it also presents a confusing and mathematically incorrect equation for the period shift. The explanation for the amplitude is correct, but the overall reasoning is misleading.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the parameters A, B, h, and D from the standard form and derives the amplitude, period, phase shift, and vertical shift accurately. The supporting equations are consistent with the stated results.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the parameters A, B, h, and D from the standard form. The derived amplitude, period, phase shift, and vertical shift are all correct and consistent with the problem statement.gpt-oss:20b: fail (error) 2026-10-15 — Sentence 3 incorrectly claims that shifting by one period changes the sign of the function; the correct statement is that y(x+π)=y(x) without a negative sign.
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.