Home›Calculus 1›Amplitude, period and phase shift›Problem 13.318

Amplitude, period and phase shift

Problem 13.318 · medium

For \( \displaystyle y = 2 - \sin{\left(4 x \right)} \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.

Write it as A·sin(B(x − h)) + D: A = 1, B = 4, h = pi/4, D = 2.
\[ \frac{2 \pi}{4} = \frac{\pi}{2} \]
Period = 2π/B.✓ Proved
\[ \left(\sin{\left(4 x \right)} - 2\right) + \left. 2 - \sin{\left(4 x \right)} \right|_{\substack{ x=x + \frac{\pi}{2} }} = 0 \]
Shifting x by one period gives the same function.✓ Proved
Phase shift h = C/B: pi/4.
\[ \left. 2 - \sin{\left(4 x \right)} \right|_{\substack{ x=\frac{3 \pi}{8} }} = 3 \]
At x = 3*pi/8 the curve is 1 from its midline y = 2: that is the amplitude.✓ Proved

Answer \( \text{amplitude } 1,\ \text{period } \frac{\pi}{2},\ \text{phase shift } \frac{\pi}{4}\text{ to the right},\ \text{vertical shift } 2 \)

Lines: 3 proved, 2 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	Not checked	—	a sentence; read, not computed
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly rewrites the function as \(A\sin(B(x-h))+D\) with \(h=\pi/4\). In fact \(y=2-\sin(4x)=2+(-1)\sin(4x)\), so \(A=-1\), \(B=4\), \(D
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly claims a phase shift of pi/4 to the right. The function y = 2 - sin(4x) is equivalent to y = 2 + sin(4x - pi), which has a p

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly claims a phase shift of pi/4 to the right. The function y = 2 - sin(4x) is equivalent to y = 2 + sin(4x - pi), which has a p
gpt-oss:20b: inconclusive 2026-10-19 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly rewrites the function as \(A\sin(B(x-h))+D\) with \(h=\pi/4\). In fact \(y=2-\sin(4x)=2+(-1)\sin(4x)\), so \(A=-1\), \(B=4\), \(D
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly claims a phase shift of pi/4 to the right. The function y = 2 - sin(4x) is equivalent to 2 + sin(4x - pi), which has a phase
gpt-oss:20b: fail (error) 2026-10-16 — The initial rewrite is incorrect: y=2−sin(4x) equals 2+(-1)·sin(4x), so A=−1, B=4, h=0, D=2. The solution’s assignment A=1 and h=π/4 misrepresents the function and leads to wrong conclusions.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies a phase shift of pi/4 to the right. The function y = 2 - sin(4x) can be written as 2 + sin(4x - pi), which corresponds to a phase shift of pi/4 to the right relative to sin(4x), but standard convention for y = A sin(Bx + C) + D defines phase shift as -C/B. Here C=0 if written as -sin(4x), or C=-pi if written as sin(4x-pi). The solution arbitrarily assigns h=pi/4 without justification from the standard form, and the direction is ambiguous without specifying the reference function. More critically, -sin(4x) is equivalent to sin(4x + pi) or sin(4x - pi). If we use sin(4(x - h)), then -sin(4x) = sin(4x + pi) = sin(4(x + pi/4)), so h = -pi/4 (left). Or -sin(4x) = sin(4x - pi) = sin(4(x - pi/4)), so h = pi/4 (right). Both are valid phase shifts, but the solution presents it as a unique derivation 'A=1, B=4, h=pi/4' from the form A·sin(B(x-h))+D without showing the transformation from -sin(4x) to sin(4(x-pi/4)). The step is logically incomplete and potentially misleading regarding the sign of A and the resulting phase.
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 \(A\sin(B(x-h))+D\) with \(A=1\), \(B=4\), \(h=\pi/4\), \(D=2\). In fact \(y=2-\sin(4x)=2+(-1)\sin(4x)\)

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.