Amplitude, period and phase shift
Problem 13.320 · medium
For \( \displaystyle y = - 3 \sin{\left(\pi x \right)} - 3 \), 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 = pi, h = 1/2, D = -3.
- \[ \frac{2 \pi}{\pi} = 2 \]Period = 2π/B.✓ Proved
- \[ \left(3 \sin{\left(\pi x \right)} + 3\right) + \left. - 3 \sin{\left(\pi x \right)} - 3 \right|_{\substack{ x=x + 2 }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\pi}{2 \pi} = \frac{1}{2} \]Phase shift h = C/B.✓ Proved
- \[ \left. - 3 \sin{\left(\pi x \right)} - 3 \right|_{\substack{ x=1 \cdot \frac{1}{2} }} = -6 \]At x = 1/2 the curve is 3 from its midline y = -3: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 3,\ \text{period } 2,\ \text{phase shift } \frac{1}{2}\text{ to the right},\ \text{vertical shift } -3 \)
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 incorrectly rewrites the function as a cosine form with A = –3, which misidentifies the amplitude and ignores the sine form. The phase shift is also incorrectly stated; the correct shift is 0 to the right, not ½. The setup and subsequent conclusions are therefore mathematically incorrect.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the phase shift as 1/2 to the right. The function y = -3 sin(pi x) - 3 has a phase shift of 0 (or can be written as 3 sin(pi(x - 1/2)) - 3, but the standard form for sine usually implies h=0 unless specified otherwise, and certainly not 1/2 to the right for the given expression). More critically, the setup in step 1 claims to write it as A*cos(B(x-h)) but uses sine parameters and values inconsistently. The stated answer claims phase shift 1/2 to the right, but -3sin(pi x) is equivalent to 3sin(pi(x - 1/2))? No, sin(pi(x-1/2)) = sin(pi x - pi/2) = -cos(pi x). So -3sin(pi x) is not a shifted cosine with h=1/2 in the standard sense without changing the function type. Actually, -sin(u) = sin(u - pi). So -3sin(pi x) = 3sin(pi x - pi) = 3sin(pi(x - 1)). So the phase shift for sine is 1. If converting to cosine: -sin(u) = cos(u + pi/2)? No, -sin(u) = cos(u + pi/2) is false. -sin(u) = cos(u + pi/2) -> cos(pi/2) = 0, -sin(0)=0. Deriv: -cos(u) vs -sin(u+pi/2)=-cos(u). Yes. So -3sin(pi x) = 3cos(pi x + pi/2) = 3cos(pi(x + 1/2)). This is a shift of 1/2 to the LEFT. The solution says RIGHT. Also, step 1 says A=-3, B=pi, h=1/2 for a cosine form, but the function is sine. The logic is garbled.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution incorrectly identifies the phase shift as 1/2 to the right. The function y = -3 sin(pi x) - 3 has a phase shift of 0 (or can be written as 3 sin(pi(x - 1/2)) - 3, but the standard form for sine usually implies h=0 unless specified otherwise, and certainly not 1/2 to the right for the given expression). More critically, the setup in step 1 claims to write it as A*cos(B(x-h)) but uses sine parameters and values inconsistently. The stated answer claims phase shift 1/2 to the right, but -3sin(pi x) is equivalent to 3sin(pi(x - 1/2))? No, sin(pi(x-1/2)) = sin(pi x - pi/2) = -cos(pi x). So -3sin(pi x) is not a shifted cosine with h=1/2 in the standard sense without changing the function type. Actually, -sin(u) = sin(u - pi). So -3sin(pi x) = 3sin(pi x - pi) = 3sin(pi(x - 1)). So the phase shift for sine is 1. If converting to cosine: -sin(u) = cos(u + pi/2)? No, -sin(u) = cos(u + pi/2) is false. -sin(u) = cos(u + pi/2) -> cos(pi/2) = 0, -sin(0)=0. Deriv: -cos(u) vs -sin(u+pi/2)=-cos(u). Yes. So -3sin(pi x) = 3cos(pi x + pi/2) = 3cos(pi(x + 1/2)). This is a shift of 1/2 to the LEFT. The solution says RIGHT. Also, step 1 says A=-3, B=pi, h=1/2 for a cosine form, but the function is sine. The logic is garbled.gpt-oss:20b: fail (error) 2026-10-19 — The solution incorrectly rewrites the function as a cosine form with A = –3, which misidentifies the amplitude and ignores the sine form. The phase shift is also incorrectly stated; the correct shift is 0 to the right, not ½. The setup and subsequent conclusions are therefore mathematically incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly identifies the phase shift as 1/2 to the right. The function y = -3 sin(pi x) - 3 can be written as 3 sin(pi x + pi) - 3, which corresponds to a phase shift of -1 (1 unit to the left), not 1/2. The step claiming h = 1/2 based on converting to cosine is algebraically inconsistent with the standard form A cos(B(x-h)) for the given sine function without accounting for the sign change properly, and the verification in step 5 checks amplitude, not phase shift.gpt-oss:20b: fail (error) 2026-10-16 — The solution incorrectly rewrites the function as a cosine with a phase shift of ½, whereas the given function is a sine with no horizontal shift. The amplitude is correctly 3, but the period calculation is correct only if B=π; however the phase shift and vertical shift are misidentified. The statement about the phase shift is false and would mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the phase shift as 1/2 to the right. The function y = -3 sin(pi x) - 3 can be written as 3 sin(pi x + pi) - 3, which corresponds to a phase shift of -1 (1 unit to the left), or equivalently 1 unit to the right if using cosine, but the derivation in step 1 claims h=1/2 for a cosine form which is inconsistent with the sine form provided without justification. Specifically, -sin(theta) = sin(theta + pi), so the shift is pi/pi = 1, not 1/2. The stated answer claims 1/2, which is incorrect.gpt-oss:20b: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly rewrites the function as a cosine form and assigns a phase shift of 1/2 to the right. The given function is \(-3\sin(\pi x)-3\);
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 2026-10-19 with SymPy 1.14.0.