Amplitude, period and phase shift
Problem 13.99 · medium
For \( \displaystyle y = 3 \sin{\left(x + \frac{\pi}{4} \right)} + 2 \), 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 = 1, h = pi/4, D = 2.
- \[ \frac{2 \pi}{1} = 2 \pi \]Period = 2π/B.✓ Proved
- \[ \left(- 3 \sin{\left(x + \frac{\pi}{4} \right)} - 2\right) + \left. 3 \sin{\left(x + \frac{\pi}{4} \right)} + 2 \right|_{\substack{ x=x + 2 \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\pi}{1 \cdot 4} = \frac{\pi}{4} \]Phase shift h = C/B.✓ Proved
- \[ \left. 3 \sin{\left(x + \frac{\pi}{4} \right)} + 2 \right|_{\substack{ x=\frac{\pi}{4} }} = 5 \]At x = pi/4 the curve is 3 from its midline y = 2: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 3,\ \text{period } 2 \pi,\ \text{phase shift } \frac{\pi}{4}\text{ to the right},\ \text{vertical shift } 2 \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. 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 misidentifies the phase shift direction: \sin(x+\pi/4) corresponds to a shift left by \pi/4, not right. The initial sentence also incorrectly rewrites the function in cosine form, which is not needed for a sine function.qwen3.6:27b-mlx: fail (error) — The solution incorrectly states the phase shift is to the right. The term (x + \pi/4) corresponds to a shift of -\pi/4, which is to the left. Additionally, step 1 incorrectly identifies the form as cosine while the function is sine, and step 5's verification is flawed because at x=\pi/4, the argument is \pi/2, yielding the maximum, but the logic conflates the location of the maximum with the definition of phase shift for the standard sine form y=Asin(B(x-h))+D where h=-\pi/4.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly states the phase shift is to the right. The term (x + \pi/4) corresponds to a shift of -\pi/4, which is to the left. Additionally, step 1 incorrectly identifies the form as cosine while the function is sine, and step 5's verification is flawed because at x=\pi/4, the argument is \pi/2, yielding the maximum, but the logic conflates the location of the maximum with the definition of phase shift for the standard sine form y=Asin(B(x-h))+D where h=-\pi/4.gpt-oss:20b: fail (misleading) 2026-10-17 — The solution misidentifies the phase shift direction: \sin(x+\pi/4) corresponds to a shift left by \pi/4, not right. The initial sentence also incorrectly rewrites the function in cosine form, which is not needed for a sine function.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The phase shift direction is incorrect. The term (x + pi/4) represents a shift of pi/4 to the left, not to the right. Additionally, step 1 incorrectly identifies the form as cosine while the function is sine, though the parameters extracted are consistent with the sine form y = A sin(B(x-h)) + D where h = -pi/4.gpt-oss:20b: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The setup incorrectly treats the function as \(A\cos(B(x-h))+D\) and assigns \(h=\pi/4\). In fact \(y=3\sin(x+\pi/4)+2=3\sin\bigl(1\cdot(x-(-\pi/4))\bigr)qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the phase shift as pi/4 to the right. For y = 3 sin(x + pi/4), the phase shift is -pi/4 (pi/4 to the left). Additionally, step 5 claims that evaluating the function at x = pi/4 demonstrates the amplitude, but f(pi/4) = 5 is the maximum value, not the amplitude itself (which is the distance from the midline, 3).gpt-oss:20b: fail (error) 2026-10-14 — The solution incorrectly states the phase shift as π/4 to the right. For y=3 sin(x+π/4)+2 the phase shift is left by π/4, not right. The conversion to a cosine form was done correctly, but the interpretation of the shift was inconsistent with the original sine form.
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.