Amplitude, period and phase shift
Problem 13.104 · easy
For \( \displaystyle y = - 2 \sin{\left(2 x \right)} - 1 \), 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 = 2, h = 0, D = -1.
- \[ \frac{2 \pi}{2} = \pi \]Period = 2π/B.✓ Proved
- \[ \left(2 \sin{\left(2 x \right)} + 1\right) + \left. - 2 \sin{\left(2 x \right)} - 1 \right|_{\substack{ x=x + \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{0}{2} = 0 \]Phase shift h = C/B.✓ Proved
- \[ \left. - 2 \sin{\left(2 x \right)} - 1 \right|_{\substack{ x=\frac{\pi}{4} }} = -3 \]At x = pi/4 the curve is 2 from its midline y = -1: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 2,\ \text{period } \pi,\ \text{phase shift } 0,\ \text{vertical shift } -1 \)
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 contains a nonsensical calculation that does not correctly demonstrate that the phase shift is zero. The expression `-(-2*sin(2*x) - 1) + Subs(-2*sin(2*x) - 1, x, x + pi) = 0` is incorrect and does not follow from the function. Consequently the conclusion about the phase shift is not properly justified.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution incorrectly identifies the amplitude as -2 (or implies A=-2 is the amplitude) in step 1, whereas amplitude is defined as the absolute value |A| = 2. While the final stated answer lists the amplitude as 2, the derivation in step 1 sets A = -2 without clarifying that amplitude is |A|, which is a conceptual error in the setup.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution incorrectly identifies the amplitude as -2 (or implies A=-2 is the amplitude) in step 1, whereas amplitude is defined as the absolute value |A| = 2. While the final stated answer lists the amplitude as 2, the derivation in step 1 sets A = -2 without clarifying that amplitude is |A|, which is a conceptual error in the setup.gpt-oss:20b: fail (error) 2026-10-17 — Sentence 3 contains a nonsensical calculation that does not correctly demonstrate that the phase shift is zero. The expression `-(-2*sin(2*x) - 1) + Subs(-2*sin(2*x) - 1, x, x + pi) = 0` is incorrect and does not follow from the function. Consequently the conclusion about the phase shift is not properly justified.qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution incorrectly identifies the amplitude as -2 (or implies A=-2 is the amplitude), whereas amplitude is defined as the absolute value |A| = 2. While the final stated answer is correct, the intermediate reasoning in step 1 and the justification in step 5 are flawed or confusingly presented, potentially teaching a student that amplitude can be negative or that the coefficient A itself is the amplitude without taking the absolute value.gpt-oss:20b: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution’s phase‑shift step uses an undefined variable C (h = C/B) and then simply states h = 0 without explaining why the shift is zero or its direction. This could confuse a student about how to determine a phase shift from the standard form.qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution incorrectly identifies the amplitude as 2, ignoring the negative sign of A. While amplitude is technically |A|, the negative sign indicates a reflection, which is often part of the 'phase' or shape description in broader contexts, but more critically, the solution fails to explicitly state that amplitude is the absolute value |A| = |-2| = 2, potentially confusing a student about the role of the negative sign. However, the bigger error is in line 1: it sets A=-2. Standard form for amplitude is usually defined with A>0 or explicitly taking absolute value. If A is allowed to be negative, the 'phase shift' calculation in line 4 (h=0) is technically correct for the form A sin(B(x-h)), but the interpretation of 'phase shift' often requires handling the negative sign as a phase shift of pi/B if one insists on A>0. The provided answer says phase shift 0, which is consistent with A=-2, h=0. But wait, is amplitude 2 correct? Yes, amplitude is always positive. Is phase shift 0 correct? If we write y = 2 sin(2x + pi) - 1, phase shift is -pi/2. If we write y = -2 sin(2x) - 1, phase shift is 0. Both are valid representations. The problem asks for 'the' phase shift. Usually, standard form implies A>0. If A>0 is required, A=2, B=2, C=pi, D=-1. Then phase shift is -pi/2. The solution uses A=-2, which is non-standard for defining 'the' phase shift uniquely without ambiguity. Most textbooks define amplitude as |A| and require A>0 for the standard form to define phase shift uniquely. By keeping A negative, the solution gives a phase shift of 0, but a student might be taught that phase shift is -C/B where the function is A sin(Bx+C) with A>0. If A is negative, it's often converted. Let's look at the 'Stated answer'. It says phase shift 0. This implies the convention used allows A<0. However, line 1 says 'Write it as... A=-2'. This is a choice. Is it wrong? Not necessarily. But line 5 says 'At x=pi/4 ... that is the amplitude'. This is a weird way to define amplitude. Amplitude is half the distance between max and min. Max is 1, min is -3. Distance 4, half is 2. The check at pi/4 gives -3, which is the minimum. The distance from midline -1 to -3 is 2. This is correct. Let's re-read the prompt's constraints. 'Find ... phase shift (with its direction)'. If the standard form is y = A sin(B(x-h)) + k, and we allow A<0, h=0 is correct. If we require A>0, h=-pi/2. The stated answer is 0. This suggests the convention A can be negative. However, there is a subtle error in Line 1. It says 'Write it as A·sin(B(x − h)) + D'. It identifies A=-2. Then Line 4 calculates 'Phase shift h = C/B'. Where did C come from? The form in Line 1 is B(x-h). There is no C. Line 4 references C, which is undefined in the context of Line 1. This is a notation error. It mixes the form A sin(Bx+C) with A sin(B(x-h)). In Line 1, it uses h. In Line 4, it uses C. This is inconsistent and confusing. Also, Line 5 is not a proof of amplitude, it's a check. But the text says 'that is the amplitude'. It's a bit loose. The main issue is the inconsistency between Line 1 (using h) and Line 4 (using C). Line 4 says 'Phase shift h = C/B'. This implies the form was A sin(Bx+C). But Line 1 said A sin(B(x-h)). If Line 1 is the setup, Line 4's formula is mismatched. If the form is A sin(B(x-h)), then h is just h. If the form is A sin(Bx+C), then h = -C/B. The solution mixes these up. Therefore, the solution is misleading/error-prone due to inconsistent notation.gpt-oss:20b: fail (error) 2026-10-14 — The solution’s third sentence attempts to justify the phase shift by shifting the function by one period, but the algebra is incorrect and does not follow from the standard form. The method for determining the phase shift is unclear and the justification for the amplitude in the last sentence is also incomplete.
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.