Amplitude, period and phase shift
Problem 13.105 · medium
For \( \displaystyle y = 4 \sin{\left(x - 1 \right)} \), 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 = 4, B = 1, h = 1, D = 0.
- \[ \frac{2 \pi}{1} = 2 \pi \]Period = 2π/B.✓ Proved
- \[ - 4 \sin{\left(x - 1 \right)} + \left. 4 \sin{\left(x - 1 \right)} \right|_{\substack{ x=x + 2 \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ 1 \cdot 1^{-1} = 1 \]Phase shift h = C/B.✓ Proved
- \[ \left. 4 \sin{\left(x - 1 \right)} \right|_{\substack{ x=1 + \frac{\pi}{2} }} = 4 \]At x = 1 + pi/2 the curve is 4 from its midline y = 0: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } 2 \pi,\ \text{phase shift } 1\text{ to the right},\ \text{vertical shift } 0 \)
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) — Sentence 3 uses a negative sign in the expression -4*sin(x-1)+Subs(...)=0, which is incorrect for demonstrating periodicity. The correct expression should involve +4*sin(x-1). This typo could mislead a student into thinking the function changes sign after a period.qwen3.6:27b-mlx: pass — The solution correctly identifies the parameters A, B, h, and D from the standard form. The algebraic checks provided in the equations are valid and support the stated values for period, phase shift, and amplitude.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the parameters A, B, h, and D from the standard form. The algebraic checks provided in the equations are valid and support the stated values for period, phase shift, and amplitude.gpt-oss:20b: fail (misleading) 2026-10-17 — Sentence 3 uses a negative sign in the expression -4*sin(x-1)+Subs(...)=0, which is incorrect for demonstrating periodicity. The correct expression should involve +4*sin(x-1). This typo could mislead a student into thinking the function changes sign after a period.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the parameters A, B, h, and D from the standard form and verifies the derived properties (period, phase shift, amplitude) with valid algebraic checks. The final answer matches the problem statement.gpt-oss:20b: fail (error) 2026-10-14 — Sentence 3 incorrectly states that shifting by one period yields the same function but the algebra shown is wrong (it uses a negative sign and an unnecessary subtraction). The reasoning for the phase shift in sentence 4 is incomplete and does not explain the direction of the shift. The solution does not clearly justify the phase shift direction or the period calculation beyond the formula.qwen3.6:27b-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 are consistent with the stated results.gpt-oss:20b: fail (error) 2026-10-14 — Sentence 3 incorrectly states that shifting by one period gives the same function but uses a negative sign and does not justify the phase‑shift direction. The solution also omits a clear statement that the phase shift is 1 unit to the right.
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.