Amplitude, period and phase shift
Problem 13.91 · medium
For \( \displaystyle y = 4 \cos{\left(4 x \right)} - 2 \), 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 = 4, h = 0, D = -2.
- \[ \frac{2 \pi}{4} = \frac{\pi}{2} \]Period = 2π/B.✓ Proved
- \[ \left(2 - 4 \cos{\left(4 x \right)}\right) + \left. 4 \cos{\left(4 x \right)} - 2 \right|_{\substack{ x=x + \frac{\pi}{2} }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{0}{4} = 0 \]Phase shift h = C/B.✓ Proved
- \[ \left. 4 \cos{\left(4 x \right)} - 2 \right|_{\substack{ x=0 }} = 2 \]At x = 0 the curve is 4 from its midline y = -2: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } \frac{\pi}{2},\ \text{phase shift } 0,\ \text{vertical shift } -2 \)
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 (style) — [domain objection, downgraded to style] The solution misapplies the phase‑shift test and misidentifies the amplitude. The calculation in step 3 is incorrect, the formula for the phase shift uses an undefined constant C, and step 5 incorrectly concludes the amplitude is 2 instead of 4.qwen3.6:27b-mlx: pass — 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.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — 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.gpt-oss:20b: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution misapplies the phase‑shift test and misidentifies the amplitude. The calculation in step 3 is incorrect, the formula for the phase shift uses an undefined constant C, and step 5 incorrectly concludes the amplitude is 2 instead of 4.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the parameters A, B, h, and D from the 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) 2026-10-14 — Sentences 3 and 5 contain incorrect reasoning: the shift test is misapplied and the amplitude is incorrectly inferred from the y‑value at x=0.qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — Step 5 incorrectly defines amplitude as the value of the function at x=0 relative to the midline, rather than the coefficient A. While the numerical result is correct for this specific function, the reasoning is false (e.g., for sin(x), f(0)=0, but amplitude is 1). Step 1 also omits the standard C parameter in the form, making the derivation in Step 4 confusing.gpt-oss:20b: fail (misleading) 2026-10-14 — Sentence 3 presents a nonsensical equation to justify the phase shift; it incorrectly applies a shift test and misrepresents the function’s periodicity, potentially misleading a student about how to determine phase shift.
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.