Amplitude, period and phase shift
Problem 13.92 · medium
For \( \displaystyle y = 4 \sin{\left(4 x + 8 \right)} + 3 \), 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 = 4, B = 4, h = -2, D = 3.Reviewed
- \[ \frac{2 \pi}{4} = \frac{\pi}{2} \]Period = 2π/B.✓ Proved
- \[ \left(- 4 \sin{\left(4 x + 8 \right)} - 3\right) + \left. 4 \sin{\left(4 x + 8 \right)} + 3 \right|_{\substack{ x=x + \frac{\pi}{2} }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ - \frac{8}{4} = -2 \]Phase shift h = C/B.✓ Proved
- \[ \left. 4 \sin{\left(4 x + 8 \right)} + 3 \right|_{\substack{ x=-2 + \frac{\pi}{8} }} = 7 \]At x = -2 + pi/8 the curve is 4 from its midline y = 3: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } \frac{\pi}{2},\ \text{phase shift } 2\text{ to the left},\ \text{vertical shift } 3 \)
Lines: 4 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the parameters A, B, h, and D by rewriting the function in the form A·sin(B(x − h)) + D. The derived amplitude, period, phase shift, and vertical shift are all correct and consistent with the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the parameters A, B, h, and D by rewriting the function in the form A·sin(B(x − h)) + D. The derived amplitude, period, phase shift, and vertical shift are all correct and consistent with the stated answer.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the parameters by rewriting the function in standard form and verifying the period, phase shift, and amplitude through algebraic checks. The interpretation of the phase shift direction (left for negative h) is correct.gpt-oss:20b: fail (error) 2026-10-14 — Sentence 5 incorrectly evaluates the function at x = -2 + π/8; it yields y = 3, not 7, so the amplitude determination is wrong.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the parameters A, B, h, and D by rewriting the function in standard form. The derived amplitude, period, phase shift, and vertical shift are all correct and consistent with the stated answer.gpt-oss:20b: fail (error) 2026-10-14 — Sentence 3 incorrectly states that shifting by one period gives the same function with a negative sign; the algebra is wrong and would mislead a student.
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 2026-10-17 with SymPy 1.14.0.