Amplitude, period and phase shift
Problem 13.271 · medium
For \( \displaystyle y = - 3 \sin{\left(4 x - 4 \right)} - 2 \), 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 = 4, h = 1, D = -2.
- \[ \frac{2 \pi}{4} = \frac{\pi}{2} \]Period = 2π/B.✓ Proved
- \[ \left(3 \sin{\left(4 x - 4 \right)} + 2\right) + \left. - 3 \sin{\left(4 x - 4 \right)} - 2 \right|_{\substack{ x=x + \frac{\pi}{2} }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{4}{4} = 1 \]Phase shift h = C/B.✓ Proved
- \[ \left. - 3 \sin{\left(4 x - 4 \right)} - 2 \right|_{\substack{ x=\frac{\pi}{8} + 1 }} = -5 \]At x = pi/8 + 1 the curve is 3 from its midline y = -2: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 3,\ \text{period } \frac{\pi}{2},\ \text{phase shift } 1\text{ to the right},\ \text{vertical shift } -2 \)
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 (misleading) — The sentence claiming h = C/B is incorrect; the correct relation is h = -C/B. The numerical result is right, but the statement could mislead a student about the formula.qwen3.6:27b-mlx: pass — The solution correctly identifies the parameters A, B, h, and D from the standard form. The derived amplitude, period, phase shift, and vertical shift match the stated answer. The supporting equations are algebraically correct and logically support the conclusions.
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. The derived amplitude, period, phase shift, and vertical shift match the stated answer. The supporting equations are algebraically correct and logically support the conclusions.gpt-oss:20b: fail (misleading) 2026-10-18 — The sentence claiming h = C/B is incorrect; the correct relation is h = -C/B. The numerical result is right, but the statement could mislead a student about the formula.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the parameters A, B, h, and D from the standard form. The derived amplitude, period, phase shift, and vertical shift are correct and consistent with the problem statement.gpt-oss:20b: fail (error) 2026-10-16 — The solution misapplies the phase‑shift formula: it uses 4/4=1 instead of C/B with C=−4, and the amplitude check uses an incorrect x‑value (π/8+1) rather than the true maximum at x=1. The setup of A, B, h, D is otherwise correct, but these errors invalidate the conclusions.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the parameters A, B, h, and D from the standard form. The derived amplitude, period, phase shift, and vertical shift match the stated answer. The supporting equations, while somewhat idiosyncratic in their verification method (e.g., checking periodicity via substitution), are algebraically correct and support the conclusions.gpt-oss:20b: pass 2026-10-15
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.