Amplitude, period and phase shift
Problem 13.269 · easy
For \( \displaystyle y = \cos{\left(2 x \right)} - 1 \), 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 = 2, h = 0, D = -1.
- \[ \frac{2 \pi}{2} = \pi \]Period = 2π/B.✓ Proved
- \[ \left(1 - \cos{\left(2 x \right)}\right) + \left. \cos{\left(2 x \right)} - 1 \right|_{\substack{ x=x + \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{0}{2} = 0 \]Phase shift h = C/B.✓ Proved
- \[ \left. \cos{\left(2 x \right)} - 1 \right|_{\substack{ x=0 }} = 0 \]At x = 0 the curve is 1 from its midline y = -1: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 1,\ \text{period } \pi,\ \text{phase shift } 0,\ \text{vertical shift } -1 \)
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 contains several incorrect or misleading statements. Sentence 3 incorrectly uses a negative sign and an undefined expression to claim that shifting by one period yields the same function; the calculation is not justified. Sentence 4 refers to a variable C that has never been defined, and the formula h=C/B is not applied correctly. Sentence 5 evaluates the function at x=0 and concludes the amplitude is 0, which contradicts the correct amplitude of 1. These errors would mislead a student and the setup is not properly justified.qwen3.6:27b-mlx: fail (misleading) — Step 5 incorrectly defines amplitude as the function value at x=0 relative to the midline, which is a coincidence for cosine but not a general definition. Step 3 is a verification of the period, not a derivation, and the phrasing is confusing.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — Step 5 incorrectly defines amplitude as the function value at x=0 relative to the midline, which is a coincidence for cosine but not a general definition. Step 3 is a verification of the period, not a derivation, and the phrasing is confusing.gpt-oss:20b: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution contains several incorrect or misleading statements. Sentence 3 incorrectly uses a negative sign and an undefined expression to claim that shifting by one period yields the same function; the calculation is not justified. Sentence 4 refers to a variable C that has never been defined, and the formula h=C/B is not applied correctly. Sentence 5 evaluates the function at x=0 and concludes the amplitude is 0, which contradicts the correct amplitude of 1. These errors would mislead a student and the setup is not properly justified.qwen3.6:27b-mlx: fail (misleading) 2026-10-16 — Step 5 incorrectly defines amplitude as the function value at x=0, which is a coincidence for this specific function and not a general definition. Amplitude is half the distance between max and min values, or |A|.gpt-oss:20b: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution’s explanation of the phase shift is unclear and uses undefined symbols (C) and a nonsensical equation. A student could be confused about why the phase shift is zero.qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] Step 5 incorrectly defines amplitude as the function value at x=0 relative to the midline, which is a coincidence for cosine but not the definition. Step 4 uses the formula h=C/B but C was not defined in the setup (Step 1 used h directly), making the derivation incoherent.gpt-oss:20b: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution’s explanation of the phase shift is unclear and uses an undefined constant C; it does not convincingly show that h=0. The step claiming the function repeats after one period is correct, but the reasoning is not properly articulated. The amplitude statement is correct, but the overall presentation could mislead a student about how to determine the 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 2026-10-18 with SymPy 1.14.0.