Amplitude, period and phase shift
Problem 13.398 · medium
For \( \displaystyle y = 4 \cos{\left(\pi x \right)} + 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.
- Write it as A·cos(B(x − h)) + D: A = -4, B = pi, h = 1, D = 2.
- \[ \frac{2 \pi}{\pi} = 2 \]Period = 2π/B.✓ Proved
- \[ \left(- 4 \cos{\left(\pi x \right)} - 2\right) + \left. 4 \cos{\left(\pi x \right)} + 2 \right|_{\substack{ x=x + 2 }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\pi}{\pi} = 1 \]Phase shift h = C/B.✓ Proved
- \[ \left. 4 \cos{\left(\pi x \right)} + 2 \right|_{\substack{ x=1 }} = -2 \]At x = 1 the curve is 4 from its midline y = 2: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } 2,\ \text{phase shift } 1\text{ to the right},\ \text{vertical shift } 2 \)
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 setup incorrectly assigns A = –4 and h = 1. The function 4 cos(πx)+2 has amplitude 4, period 2, no phase shift (h = 0), and vertical shift 2. The later sentences also misinterpret the value at x = 1 as the amplitude.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the phase shift as 1 to the right. The function y = 4 cos(pi x) + 2 has a phase shift of 0. The solver likely confused the first zero-crossing or minimum with the phase shift, or incorrectly manipulated the sign of A to force a shift.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution incorrectly identifies the phase shift as 1 to the right. The function y = 4 cos(pi x) + 2 has a phase shift of 0. The solver likely confused the first zero-crossing or minimum with the phase shift, or incorrectly manipulated the sign of A to force a shift.gpt-oss:20b: fail (error) 2026-10-20 — The setup incorrectly assigns A = –4 and h = 1. The function 4 cos(πx)+2 has amplitude 4, period 2, no phase shift (h = 0), and vertical shift 2. The later sentences also misinterpret the value at x = 1 as the amplitude.qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution incorrectly identifies the phase shift as 1 to the right. The function y = 4 cos(πx) + 2 has no horizontal shift (h=0). The step claiming h=1 likely confuses the phase shift with the location of a specific point (like a minimum) or incorrectly factors the cosine argument. Additionally, Step 1 sets A=-4, which contradicts the standard positive amplitude definition used in the final answer, and Step 5's logic for amplitude is convoluted and relies on the incorrect x=1 evaluation.gpt-oss:20b: fail (error) 2026-10-19 — The solution misidentifies the amplitude and phase shift. It assigns A = –4 (instead of |A| = 4) and h = 1, whereas the correct form is 4 cos(πx)+2 with h = 0. The amplitude calculation using x = 1 is also incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution incorrectly identifies the phase shift as 1 to the right. The function y = 4 cos(pi x) + 2 has a phase shift of 0. The step claiming h=1 is derived from an incorrect algebraic manipulation or misinterpretation of the standard form, leading to a false conclusion about the phase shift.gpt-oss:20b: fail (error) 2026-10-19 — The solution misidentifies the amplitude sign, the phase shift, and the shift parameter. It incorrectly rewrites the function as A·cos(B(x−h))+D with A = −4 and h = 1, whereas the correct form is 4·cos(π(x−0))+2, giving a phase shift of 0. The subsequent calculations for period, phase shift, and amplitude are based on these wrong parameters and therefore are incorrect.
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.