Amplitude, period and phase shift
Problem 13.317 · medium
For \( \displaystyle y = 3 - 3 \sin{\left(2 x - 2 \right)} \), 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 = -3, B = 2, h = 1, D = 3.
- \[ \frac{2 \pi}{2} = \pi \]Period = 2π/B.✓ Proved
- \[ \left(3 \sin{\left(2 x - 2 \right)} - 3\right) + \left. 3 - 3 \sin{\left(2 x - 2 \right)} \right|_{\substack{ x=x + \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{2}{2} = 1 \]Phase shift h = C/B.✓ Proved
- \[ \left. 3 - 3 \sin{\left(2 x - 2 \right)} \right|_{\substack{ x=\frac{\pi}{4} + 1 }} = 0 \]At x = pi/4 + 1 the curve is 3 from its midline y = 3: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 3,\ \text{period } \pi,\ \text{phase shift } 1\text{ to the right},\ \text{vertical shift } 3 \)
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 misidentifies the amplitude sign, incorrectly applies the period shift test, and incorrectly evaluates the function at the claimed phase‑shift point, leading to false statements about the function’s values.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution incorrectly identifies the amplitude as -3 in step 1. Amplitude is defined as the absolute value |A|, so it must be 3. While the final stated answer lists amplitude 3, the intermediate reasoning in step 1 is mathematically incorrect and misleading.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] The solution incorrectly identifies the amplitude as -3 in step 1. Amplitude is defined as the absolute value |A|, so it must be 3. While the final stated answer lists amplitude 3, the intermediate reasoning in step 1 is mathematically incorrect and misleading.gpt-oss:20b: fail (error) 2026-10-19 — The solution misidentifies the amplitude sign, incorrectly applies the period shift test, and incorrectly evaluates the function at the claimed phase‑shift point, leading to false statements about the function’s values.qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 5 is mathematically incorrect: substituting x = pi/4 + 1 into the function yields 3 - 3*sin(pi/2) = 0, not the amplitude. The amplitude is the coefficient magnitude |A|, not the function value at a specific point. Additionally, Step 1 identifies A = -3, but the final answer lists amplitude as 3 without explicitly stating that amplitude is |A|, which is a minor omission but Step 5 is a hard error.gpt-oss:20b: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution misstates several facts: the phase‑shift calculation uses an undefined constant C, the period‑shift test is written incorrectly, and the evalqwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution incorrectly identifies the amplitude as -3 in step 1, whereas amplitude is defined as the absolute value |A| = 3. Additionally, step 5 contains a factual error: substituting x = pi/4 + 1 into the function yields 3 - 3*sin(pi/2) = 0, not 3, so the curve is at its minimum, not 3 units from the midline in the positive direction (though the magnitude is correct, the reasoning is flawed).gpt-oss:20b: fail (misleading) 2026-10-15 — The solution states A = –3 and later refers to the amplitude as 3, but it never explicitly identifies the amplitude as |A| = 3. This could mislead a student into thinking the amplitude is –3, which is incorrect. The rest of the setup and calculations are correct, but the wording about amplitude is misleading.
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 2026-10-19 with SymPy 1.14.0.