Amplitude, period and phase shift
Problem 13.279 · easy
For \( \displaystyle y = 2 \sin{\left(x \right)} - 3 \), 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 = 2, B = 1, h = 0, D = -3.
- \[ \frac{2 \pi}{1} = 2 \pi \]Period = 2π/B.✓ Proved
- \[ \left(3 - 2 \sin{\left(x \right)}\right) + \left. 2 \sin{\left(x \right)} - 3 \right|_{\substack{ x=x + 2 \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{0}{1} = 0 \]Phase shift h = C/B.✓ Proved
- \[ \left. 2 \sin{\left(x \right)} - 3 \right|_{\substack{ x=\frac{\pi}{2} }} = -1 \]At x = pi/2 the curve is 2 from its midline y = -3: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 2,\ \text{period } 2 \pi,\ \text{phase shift } 0,\ \text{vertical shift } -3 \)
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 (style) — [domain objection, downgraded to style] The solution contains several incorrect or misleading statements: sentence 3 presents a nonsensical equation to claim periodicity; sentence 4 uses an undefined variable C to compute the phase shift; sentence 5 misstates the reasoning for amplitude. These errors would confuse a student and do not correctly justify the requested properties.qwen3.6:27b-mlx: pass — The solution correctly identifies the parameters A, B, h, and D from the standard form and derives the requested properties. The supporting equations are algebraically correct and consistent with the stated results.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the parameters A, B, h, and D from the standard form and derives the requested properties. The supporting equations are algebraically correct and consistent with the stated results.gpt-oss:20b: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution contains several incorrect or misleading statements: sentence 3 presents a nonsensical equation to claim periodicity; sentence 4 uses an undefined variable C to compute the phase shift; sentence 5 misstates the reasoning for amplitude. These errors would confuse a student and do not correctly justify the requested properties.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 requested properties. The supporting equations, while somewhat idiosyncratic in their verification methods (e.g., checking periodicity via substitution), are mathematically valid and support the conclusions.gpt-oss:20b: fail (error) 2026-10-16 — The solution misapplies the phase‑shift test (step 3) and incorrectly identifies the amplitude (step 5).qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the parameters A, B, h, and D from the standard form and derives the requested properties. The supporting equations, while somewhat redundant or oddly phrased (e.g., checking periodicity via substitution), are mathematically valid and do not lead to false conclusions.gpt-oss:20b: fail (error) 2026-10-15 — Sentence 3 incorrectly attempts to determine the phase shift by shifting the function by one period; this does not establish the phase shift and the algebraic manipulation is nonsensical. The subsequent sentence 4 also incorrectly applies the formula for phase shift without identifying the horizontal translation constant. The solution therefore contains a faulty step in the reasoning about 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.