Amplitude, period and phase shift
Problem 13.400 · medium
For \( \displaystyle y = - 3 \sin{\left(\frac{x}{2} \right)} \), 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 = -3, B = 1/2, h = 0, D = 0.
- \[ \frac{2 \pi}{1 \cdot \frac{1}{2}} = 4 \pi \]Period = 2π/B.✓ Proved
- \[ 3 \sin{\left(\frac{x}{2} \right)} + \left. - 3 \sin{\left(\frac{x}{2} \right)} \right|_{\substack{ x=x + 4 \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{0}{1 \cdot \frac{1}{2}} = 0 \]Phase shift h = C/B.✓ Proved
- \[ \left. - 3 \sin{\left(\frac{x}{2} \right)} \right|_{\substack{ x=\pi }} = -3 \]At x = pi the curve is 3 from its midline y = 0: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 3,\ \text{period } 4 \pi,\ \text{phase shift } 0,\ \text{vertical shift } 0 \)
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 (error) — Sentence 3 incorrectly asserts that shifting the function by one period yields zero; the substitution expression is not equal to zero and does not demonstrate periodicity. The statement is mathematically incorrect and could mislead a student.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the amplitude as 3 while simultaneously identifying A = -3. Although the magnitude is correct, the negative sign
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-20 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the amplitude as 3 while simultaneously identifying A = -3. Although the magnitude is correct, the negative signgpt-oss:20b: fail (error) 2026-10-20 — Sentence 3 incorrectly asserts that shifting the function by one period yields zero; the substitution expression is not equal to zero and does not demonstrate periodicity. The statement is mathematically incorrect and could mislead a student.qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to correctly identify the amplitude. While it correctly identifies A = -3, it does not explicitly state that amplitude is |A| = 3gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The solution incorrectly defines amplitude as 3, ignoring the negative sign of A=-3 which implies a reflection. While amplitude is technically the absolute value, the setup A=-3 should lead to a discussion of reflection or explicitly stating amplitude = |A|. More critically, step 5's justification is flawed: it claims the value -3 is '3 from its midline', which is true, but it fails to explicitly connect this to the definition of amplitude as the maximum displacement, and the negative sign in A indicates the wave starts by going down, not up. The primary issue is that the solution presents A=-3 but concludes amplitude is 3 without explicitly stating the absolute value operation, which can confuse students about the role of the negative sign (reflection vs amplitude magnitude). However, the most significant error is in the interpretation of the phase shift formula in step 4: it uses h = C/B but there is no C in the original equation y = -3 sin(x/2), so C=0 is implied but not stated, making the derivation opaque. Actually, the standard form is often y = A sin(Bx - C) + D or y = A sin(B(x-h)) + D. The solution mixes these. Step 1 says A sin(B(x-h)), so h=0. Step 4 says h=C/B. This is inconsistent notation without defining C. If the form is B(x-h), then h is the phase shift. If the form is Bx - C, then phase shift is C/B. The solution uses the first form in step 1 but the second form's logic in step 4 without defining C. This is misleading.gpt-oss:20b: fail (misleading) 2026-10-19 — Sentence 3 claims that shifting x by one period gives the value 0, which is incorrect; the function remains unchanged, not zero. This could mislead a student into thinking the function evaluates to zero after a period shift.
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 2026-10-20 with SymPy 1.14.0.