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Amplitude, period and phase shift

Problem 13.314 · medium

For \( \displaystyle y = 3 \sin{\left(\frac{x}{2} - \frac{1}{2} \right)} - 1 \), 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 = 3, B = 1/2, h = 1, D = -1.
Reviewed
\[ \frac{2 \pi}{1 \cdot \frac{1}{2}} = 4 \pi \]
Period = 2π/B.✓ Proved
\[ \left(1 - 3 \sin{\left(\frac{x}{2} - 1 \cdot \frac{1}{2} \right)}\right) + \left. 3 \sin{\left(\frac{x}{2} - 1 \cdot \frac{1}{2} \right)} - 1 \right|_{\substack{ x=x + 4 \pi }} = 0 \]
Shifting x by one period gives the same function.✓ Proved
\[ 1 \frac{1}{1 \cdot 2 \cdot \frac{1}{2}} = 1 \]
Phase shift h = C/B.✓ Proved
\[ \left. 3 \sin{\left(\frac{x}{2} - 1 \cdot \frac{1}{2} \right)} - 1 \right|_{\substack{ x=1 + \pi }} = 2 \]
At x = 1 + pi the curve is 3 from its midline y = -1: that is the amplitude.✓ Proved

Answer \( \text{amplitude } 3,\ \text{period } 4 \pi,\ \text{phase shift } 1\text{ to the right},\ \text{vertical shift } -1 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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: pass
qwen3.6:27b-mlx: pass — 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 correct and consistent with the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — 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 correct and consistent with the stated answer.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — 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 correct and consistent with the standard definitions.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — 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 match the stated answer and are mathematically sound.
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 2026-10-19 with SymPy 1.14.0.