Amplitude, period and phase shift
Problem 13.96 · medium
For \( \displaystyle y = - 3 \cos{\left(\frac{\pi x}{2} \right)} - 1 \), 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 = pi/2, h = -1, D = -1.
- \[ \frac{2 \pi}{\frac{1}{2} \pi} = 4 \]Period = 2π/B.✓ Proved
- \[ \left(3 \cos{\left(\frac{\pi x}{2} \right)} + 1\right) + \left. - 3 \cos{\left(\frac{\pi x}{2} \right)} - 1 \right|_{\substack{ x=x + 4 }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\left(-1\right) \pi}{2 \frac{\pi}{2}} = -1 \]Phase shift h = C/B.✓ Proved
- \[ \left. - 3 \cos{\left(\frac{\pi x}{2} \right)} - 1 \right|_{\substack{ x=0 }} = -4 \]At x = 0 the curve is 3 from its midline y = -1: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 3,\ \text{period } 4,\ \text{phase shift } 1\text{ to the left},\ \text{vertical shift } -1 \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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) — The solution misidentifies the phase shift: the function has no horizontal shift (C=0), yet the text claims a shift of 1 to the left. The algebraic step used to justify this is incorrect and would mislead a student.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the phase shift as 1 to the left (h=-1) when the function y = -3 cos(pi x/2) - 1 has no horizontal shift (h=0). The equation in step 4 calculates -1 but mislabels it as the phase shift h, likely confusing the coefficient of x or the sign of the cosine term with a horizontal translation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly identifies the phase shift as 1 to the left (h=-1) when the function y = -3 cos(pi x/2) - 1 has no horizontal shift (h=0). The equation in step 4 calculates -1 but mislabels it as the phase shift h, likely confusing the coefficient of x or the sign of the cosine term with a horizontal translation.gpt-oss:20b: fail (error) 2026-10-17 — The solution misidentifies the phase shift: the function has no horizontal shift (C=0), yet the text claims a shift of 1 to the left. The algebraic step used to justify this is incorrect and would mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the phase shift as 1 to the left (h = -1) when the function y = -3 cos(pi x / 2) - 1 has no horizontal shift (h = 0). The negative sign is part of the amplitude/reflection, not a phase shift. Additionally, the setup in step 1 claims the form is A*sin(B(x-h)) but uses cosine values, creating a contradiction.gpt-oss:20b: fail (error) 2026-10-14 — The setup incorrectly rewrites the cosine function as a sine form and assigns the phase shift h = –1. The correct phase shift is 2 units to the left (h = 2), derived from B h = π. The amplitude should be |A| = 3, not A = –3.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the phase shift as 1 to the left (h=-1) when the function y = -3 cos(pi x/2) - 1 has no horizontal shift (h=0). The phase shift calculation in step 4 is mathematically nonsensical and does not correspond to the standard form conversion.gpt-oss:20b: fail (error) 2026-10-14 — The solution incorrectly rewrites the function as A·sin(B(x−h))+D with A = −3, h = −1, and D = −1. This misidentifies the amplitude (it should be 3), the phase shift, and the vertical shift. The subsequent calculations based on these wrong parameters are therefore invalid.
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.