Amplitude, period and phase shift
Problem 13.475 · medium
For \( \displaystyle y = 3 \sin{\left(2 x \right)} + 2 \), 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 = 3, B = 2, h = pi/4, D = 2.
- \[ \frac{2 \pi}{2} = \pi \]Period = 2π/B.✓ Proved
- \[ \left(- 3 \sin{\left(2 x \right)} - 2\right) + \left. 3 \sin{\left(2 x \right)} + 2 \right|_{\substack{ x=x + \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\pi}{2 \cdot 2} = \frac{\pi}{4} \]Phase shift h = C/B.✓ Proved
- \[ \left. 3 \sin{\left(2 x \right)} + 2 \right|_{\substack{ x=\frac{\pi}{4} }} = 5 \]At x = pi/4 the curve is 3 from its midline y = 2: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 3,\ \text{period } \pi,\ \text{phase shift } \frac{\pi}{4}\text{ to the right},\ \text{vertical shift } 2 \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. 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) — The solution incorrectly models the function as a cosine with a phase shift of π/4. In fact y=3sin(2x)+2 has no horizontal shift (phase shift 0). The equations claiming a shift and the subsequent calculations are mathematically invalid.qwen3.6:27b-mlx: fail (error) — The solution claims a phase shift of pi/4 to the right, but the function y = 3sin(2x) + 2 has no horizontal phase shift (h=0). The value pi/4 is the location of the first maximum, not the phase shift. Additionally, the setup in step 1 incorrectly asserts h=pi/4.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution claims a phase shift of pi/4 to the right, but the function y = 3sin(2x) + 2 has no horizontal phase shift (h=0). The value pi/4 is the location of the first maximum, not the phase shift. Additionally, the setup in step 1 incorrectly asserts h=pi/4.gpt-oss:20b: fail (error) 2026-11-08 — The solution incorrectly models the function as a cosine with a phase shift of π/4. In fact y=3sin(2x)+2 has no horizontal shift (phase shift 0). The equations claiming a shift and the subsequent calculations are mathematically invalid.qwen3.6:27b-mlx: fail (error) 2026-11-07 — The solution incorrectly identifies a phase shift of pi/4 for the function y = 3 sin(2x) + 2, which has a phase shift of 0. It falsely claims the function can be written with h = pi/4 in the form A·cos(B(x-h)) + D, and incorrectly validates this by checking the amplitude at x=pi/4 rather than verifying the phase shift itself.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: fail (error) 2026-11-07 — The solution incorrectly claims the phase shift is pi/4 to the right. The function y = 3sin(2x) + 2 has a phase shift of 0 (or multiples of the period), as there is no horizontal translation inside the sine argument. The value pi/4 is the location of the first maximum, not the phase shift. Additionally, Step 1 attempts to rewrite a sine function as a cosine with a specific shift but fails to account for the identity sin(theta) = cos(theta - pi/2), leading to an incorrect h value for the cosine form as well.gpt-oss:20b: fail (error) 2026-11-07 — The solution contains several incorrect or unclear statements. Sentence 1 correctly identifies the cosine form, but sentence 3 uses a nonsensical expression to claim periodicity, sentence 4 incorrectly applies a formula for phase shift without defining the needed constant, and sentence 5 misstates the reasoning for amplitude. These errors would mislead a student.
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 2026-11-08 with SymPy 1.14.0.