Amplitude, period and phase shift
Problem 13.98 · medium
For \( \displaystyle y = 2 \sin{\left(3 x \right)} + 1 \), 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 = 3, h = 0, D = 1.
- Period = 2π/B: 2*pi/3.
- \[ \left(- 2 \sin{\left(3 x \right)} - 1\right) + \left. 2 \sin{\left(3 x \right)} + 1 \right|_{\substack{ x=x + \frac{2 \pi}{3} }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{0}{3} = 0 \]Phase shift h = C/B.✓ Proved
- \[ \left. 2 \sin{\left(3 x \right)} + 1 \right|_{\substack{ x=\frac{\pi}{6} }} = 3 \]At x = pi/6 the curve is 2 from its midline y = 1: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 2,\ \text{period } \frac{2 \pi}{3},\ \text{phase shift } 0,\ \text{vertical shift } 1 \)
Lines: 3 proved, 2 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 | Not checked | — | a sentence; read, not computed |
| 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) — Sentence 3 incorrectly sets up the period test with a negative sign and an ill‑formed equation; sentence 5 misinterprets the value at x=π/6 as the amplitude instead of using the coefficient 2. Both statements would mislead a student.qwen3.6:27b-mlx: fail (misleading) — The solution incorrectly identifies the amplitude by evaluating the function at a specific point (x = pi/6) rather than extracting the coefficient A. While the numerical result is correct for this specific function, the method is not general (e.g., it fails for cosine or shifted sine waves) and teaches a false procedure for finding amplitude.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution incorrectly identifies the amplitude by evaluating the function at a specific point (x = pi/6) rather than extracting the coefficient A. While the numerical result is correct for this specific function, the method is not general (e.g., it fails for cosine or shifted sine waves) and teaches a false procedure for finding amplitude.gpt-oss:20b: fail (error) 2026-10-17 — Sentence 3 incorrectly sets up the period test with a negative sign and an ill‑formed equation; sentence 5 misinterprets the value at x=π/6 as the amplitude instead of using the coefficient 2. Both statements would mislead a student.qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The solution incorrectly identifies the amplitude by evaluating the function at a specific point (x=pi/6) rather than using the coefficient A. While the numerical result is correct for this specific function, the method is not a general definition of amplitude and would fail for functions with phase shifts or different frequencies.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the parameters A, B, h, and D from the standard form and derives the correct amplitude, period, phase shift, and vertical shift. The supporting equations, while slightly informal in their commentary, are mathematically consistent with the results.gpt-oss:20b: fail (style) 2026-10-14 — [domain objection, downgraded to style] Sentence 4 incorrectly introduces an undefined variable C and claims h = C/B; the phase shift is already determined to be 0, so this step misleads the reader.
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.