Amplitude, period and phase shift
Problem 13.396 · medium
For \( \displaystyle y = \cos{\left(4 x \right)} \), 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 = 1, B = 4, h = 0, D = 0.
- \[ \frac{2 \pi}{4} = \frac{\pi}{2} \]Period = 2π/B.✓ Proved
- \[ - \cos{\left(4 x \right)} + \left. \cos{\left(4 x \right)} \right|_{\substack{ x=x + \frac{\pi}{2} }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{0}{4} = 0 \]Phase shift h = C/B.✓ Proved
- \[ \left. \cos{\left(4 x \right)} \right|_{\substack{ x=0 }} = 1 \]At x = 0 the curve is 1 from its midline y = 0: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 1,\ \text{period } \frac{\pi}{2},\ \text{phase shift } 0,\ \text{vertical shift } 0 \)
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 (error) — Sentence 3 incorrectly claims that shifting by one period yields the same function, but the algebra shown is nonsensical and does not establish the phase shift. The conclusion that the phase shift is zero is correct, but the justification is invalid.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] Step 5 incorrectly defines amplitude as the function value at x=0 relative to the midline. While true for this specific function, amplitude is defined as |A|, not f(0) - D, which fails for functions with non-zero phase shifts (e.g., y = cos(x - pi/2) has amplitude 1 but f(0)=0).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-20 — [domain objection, downgraded to style] Step 5 incorrectly defines amplitude as the function value at x=0 relative to the midline. While true for this specific function, amplitude is defined as |A|, not f(0) - D, which fails for functions with non-zero phase shifts (e.g., y = cos(x - pi/2) has amplitude 1 but f(0)=0).gpt-oss:20b: fail (error) 2026-10-20 — Sentence 3 incorrectly claims that shifting by one period yields the same function, but the algebra shown is nonsensical and does not establish the phase shift. The conclusion that the phase shift is zero is correct, but the justification is invalid.qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — Step 5 incorrectly defines amplitude as the function value at x=0. Amplitude is the coefficient A, not f(0); for y=sin(4x), f(0)=0 but amplitude is 1.gpt-oss:20b: fail (style) 2026-10-19 — [domain objection, downgraded to style] Sentences 3 and 4 contain incorrect reasoning: the shift test is misapplied and the phase‑shift formula is used with an undefined constant. The solution does not correctly justify the phase shift and therefore is not sound.qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] Step 5 incorrectly defines amplitude as the function value at x=0 relative to the midline. While true for this specific function, amplitude is defined as the coefficient |A|, not the value at a specific point (which fails for phase-shifted cosines like cos(x - pi/2) = sin(x) where f(0)=0 but amplitude is 1).gpt-oss:20b: fail (error) 2026-10-19 — Sentence 3 incorrectly asserts that shifting by one period yields a zero expression; the algebra shown is nonsensical and does not support the claim that the function repeats after a period.
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 2026-10-20 with SymPy 1.14.0.