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Amplitude, period and phase shift

Problem 13.103 · medium

For \( \displaystyle y = 2 - 2 \sin{\left(x + \frac{\pi}{3} \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.
  1. Write it as A·sin(B(x − h)) + D: A = -2, B = 1, h = -pi/3, D = 2.
  2. \[ \frac{2 \pi}{1} = 2 \pi \]
    Period = 2π/B.✓ Proved
  3. \[ \left(2 \sin{\left(x + \frac{\pi}{3} \right)} - 2\right) + \left. 2 - 2 \sin{\left(x + \frac{\pi}{3} \right)} \right|_{\substack{ x=x + 2 \pi }} = 0 \]
    Shifting x by one period gives the same function.✓ Proved
  4. \[ \frac{\left(-1\right) \pi}{1 \cdot 3} = - \frac{\pi}{3} \]
    Phase shift h = C/B.✓ Proved
  5. \[ \left. 2 - 2 \sin{\left(x + \frac{\pi}{3} \right)} \right|_{\substack{ x=\frac{\pi}{6} }} = 0 \]
    At x = pi/6 the curve is 2 from its midline y = 2: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 2,\ \text{period } 2 \pi,\ \text{phase shift } \frac{\pi}{3}\text{ to the left},\ \text{vertical shift } 2 \)

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
LineStatusChecked byDetail
1Not checked—a sentence; read, not computed
2✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
3✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
4✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
5✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
answer, a second way✓ Checked independentlysympy 1.14.0 + mpmath 1.3.0max 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The solution’s sentences are confusing and incorrectly justify the amplitude. It states that evaluating at \(x=\pi/6\) gives 0, yet claims that shows
  • qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the phase shift as -pi/3 (or pi/3 to the left) in step 1 and step 4. For y = A sin(B(x - h)) + D, the term (x +
Every verdict on record (6)
  • qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the phase shift as -pi/3 (or pi/3 to the left) in step 1 and step 4. For y = A sin(B(x - h)) + D, the term (x +
  • gpt-oss:20b: inconclusive 2026-10-17 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The solution’s sentences are confusing and incorrectly justify the amplitude. It states that evaluating at \(x=\pi/6\) gives 0, yet claims that shows
  • qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution incorrectly identifies the phase shift as pi/3 to the left. The term (x + pi/3) corresponds to a shift of -pi/3, which is pi/3 to the left, but the standard form A sin(B(x-h)) + D implies h = -pi/3. While the final answer states 'pi/3 to the left', the intermediate step 4 calculates h = -pi/3 and step 1 identifies h = -pi/3, but the final stated answer in the problem description says 'phase shift pi/3 to the left'. Wait, let's look closer. The problem asks for phase shift with direction. The solution finds h = -pi/3. A shift of h means the graph is shifted by h. So x -> x - h. If h = -pi/3, x -> x + pi/3. This is a shift to the left by pi/3. The stated answer says 'phase shift pi/3 to the left'. This is correct. However, step 5 is nonsensical. It claims that at x=pi/6, the curve is 2 from its midline, proving amplitude. Let's check: y(pi/6) = 2 - 2sin(pi/6 + pi/3) = 2 - 2sin(pi/2) = 2 - 2(1) = 0. The midline is y=2. The distance is |0 - 2| = 2. This is correct. But is it a valid proof of amplitude? It shows one point is at the extremum. It doesn't prove it's the maximum deviation everywhere, but for sine/cosine it's implied. The main issue is usually the sign convention. Let's re-read the prompt's stated answer: 'phase shift pi/3 to the left'. The solution steps: Step 1 says h = -pi/3. Step 4 calculates h = -pi/3. The text says 'phase shift pi/3 to the left'. This is consistent. Is there an error? Let's check the amplitude sign. A = -2. Amplitude is |A| = 2. Correct. Period 2pi. Correct. Vertical shift 2. Correct. Phase shift: sin(x + pi/3) = sin(x - (-pi/3)). Shift is -pi/3, i.e., left by pi/3. Correct. Why would it fail? Maybe the 'sentence' in step 1 is misleading? 'Write it as A·sin(B(x − h)) + D: A = -2, B = 1, h = -pi/3, D = 2.' This is a correct identification. Step 5 is weirdly phrased but mathematically true. Is there a subtle error? Ah, look at step 1 again. It defines the form. Then step 4 uses 'Phase shift h = C/B'. This assumes the form A sin(Bx + C) + D where phase shift is -C/B or C/B depending on definition. If form is B(x-h), shift is h. If form is Bx+C, shift is -C/B. Here C = pi/3. -C/B = -pi/3. The solution says h = -pi/3. This is consistent. The stated answer says 'pi/3 to the left'. This is consistent with h = -pi/3. I don't see an error. Wait, let me look at the 'Stated answer' vs 'Solution'. The solution steps don't explicitly state the final answer, they just derive components. The prompt asks me to review the solution. Is the solution complete? It derives A, B, h, D. It doesn't explicitly combine them into the final sentence, but the components are correct. Let's look for a 'misleading' or 'style' issue. Step 5 is very poor style/logic. 'At x = pi/6 the curve is 2 from its midline... that is the amplitude.' This is an example, not a derivation. It's not a rigorous proof, but in this context, it's likely acceptable. However, step 1 says 'A = -2'. Amplitude is defined as |A|. The solution doesn't explicitly take the absolute value, but the stated answer has 'amplitude 2'. If the solution implies A is the amplitude, that's wrong (A can be negative). But it lists A=-2. Then the stated answer says amplitude 2. The connection is missing in the text. This is a 'misleading' gap. A student might think Amplitude = A = -2. The solution fails to state that Amplitude = |A|. This is a significant pedagogical error.
  • gpt-oss:20b: fail (error) 2026-10-14 — The solution misidentifies the amplitude as A (negative) and incorrectly applies the phase‑shift formula. The phase shift is h = –π/3, so the graph shifts left by π/3, but the text uses a confusing C/B notation. Equation 3 is nonsensical (it attempts to show periodicity with a zero expression). Sentence 5 incorrectly claims the curve is 2 units from the midline at x=π/6 without explaining that this is the amplitude; the reasoning is unclear and could mislead a student.
  • qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the phase shift as -pi/3 (left) when the term (x + pi/3) corresponds to a shift of -pi/3, which is pi/3 to the le
  • gpt-oss:20b: pass 2026-10-14

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.

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