Amplitude, period and phase shift
Problem 13.278 · medium
For \( \displaystyle y = - 4 \sin{\left(\frac{\pi x}{2} + \frac{\pi}{4} \right)} - 1 \), 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/2, h = 1/2, D = -1.
- \[ \frac{2 \pi}{\frac{1}{2} \pi} = 4 \]Period = 2π/B.✓ Proved
- \[ \left(4 \sin{\left(\frac{\pi x}{2} + \frac{\pi}{4} \right)} + 1\right) + \left. - 4 \sin{\left(\frac{\pi x}{2} + \frac{\pi}{4} \right)} - 1 \right|_{\substack{ x=x + 4 }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\pi}{4 \frac{\pi}{2}} = \frac{1}{2} \]Phase shift h = C/B.✓ Proved
- \[ \left. - 4 \sin{\left(\frac{\pi x}{2} + \frac{\pi}{4} \right)} - 1 \right|_{\substack{ x=1 \cdot \frac{1}{2} }} = -5 \]At x = 1/2 the curve is 4 from its midline y = -1: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } 4,\ \text{phase shift } \frac{1}{2}\text{ to the right},\ \text{vertical shift } -1 \)
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 (misleading) — The solution states a phase shift of 1/2 to the right, but for the given sine form the shift is actually 1/2 to the left. This would mislead a student about the direction of the phase shift.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the phase shift as 1/2 to the right. For y = -4 sin(pi/2(x + 1/2)) - 1, the phase shift is -1/2 (1/2 to the left). The calculation C/B = (pi/4)/(pi/2) = 1/2 gives the magnitude, but the sign is wrong because the argument is (pi/2)x + pi/4, which corresponds to a shift of -C/B. Additionally, step 1 claims to write it as A*cos(B(x-h)), but the subsequent steps use sine properties and the original sine form, creating a contradiction in the stated method.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly identifies the phase shift as 1/2 to the right. For y = -4 sin(pi/2(x + 1/2)) - 1, the phase shift is -1/2 (1/2 to the left). The calculation C/B = (pi/4)/(pi/2) = 1/2 gives the magnitude, but the sign is wrong because the argument is (pi/2)x + pi/4, which corresponds to a shift of -C/B. Additionally, step 1 claims to write it as A*cos(B(x-h)), but the subsequent steps use sine properties and the original sine form, creating a contradiction in the stated method.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution states a phase shift of 1/2 to the right, but for the given sine form the shift is actually 1/2 to the left. This would mislead a student about the direction of the phase shift.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly identifies the phase shift as 1/2 to the right. For y = -4 sin(pi x/2 + pi/4), the phase shift is -1/2 (1/2 to the left). The calculation C/B = (pi/4)/(pi/2) = 1/2 gives the magnitude, but the sign is wrong because the argument is (Bx + C), implying a shift of -C/B. Additionally, step 5 claims the value at x=1/2 demonstrates the amplitude, but f(1/2) = -5, which is a minimum, not a point that directly defines amplitude without referencing the midline, and x=1/2 is not the phase shift location.gpt-oss:20b: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution mis‑applies the sine form: it treats the function as \(A\cos(B(x-h))+D\) and sets \(A=-4\), which is incorrect because amplitude is the absolqwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the phase shift as 1/2 to the right. For y = A sin(Bx + C) + D, the phase shift is -C/B. Here, -C/B = -(pi/4)/(pi/2) = -1/2, which corresponds to a shift of 1/2 to the left. The calculation in line 4 computes C/B but ignores the negative sign required for the shift direction relative to the standard sine function, leading to the wrong direction.gpt-oss:20b: fail (error) 2026-10-15 — The solution incorrectly rewrites the function as a cosine with a negative amplitude and asserts a phase shift of +½ to the right. The correct form is y = 4 sin(πx/2 + π/4) – 1, giving amplitude 4, period 4, phase shift –¼ (i.e., ¼ to the left), and vertical shift –1. The sign of the amplitude and the direction of the phase shift are both wrong, which 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 2026-10-18 with SymPy 1.14.0.