Amplitude, period and phase shift
Problem 13.391 · medium
For \( \displaystyle y = 4 \cos{\left(\pi x \right)} - 2 \), 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 = -4, B = pi, h = 1/2, D = -2.
- \[ \frac{2 \pi}{\pi} = 2 \]Period = 2π/B.✓ Proved
- \[ \left(2 - 4 \cos{\left(\pi x \right)}\right) + \left. 4 \cos{\left(\pi x \right)} - 2 \right|_{\substack{ x=x + 2 }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{\pi}{2 \pi} = \frac{1}{2} \]Phase shift h = C/B.✓ Proved
- \[ \left. 4 \cos{\left(\pi x \right)} - 2 \right|_{\substack{ x=1 }} = -6 \]At x = 1 the curve is 4 from its midline y = -2: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } 2,\ \text{phase shift } \frac{1}{2}\text{ to the right},\ \text{vertical shift } -2 \)
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 (error) — The solution incorrectly rewrites the function as A·sin(B(x−h))+D with A = –4 and h = ½. The amplitude of 4 cos(πx)–2 is 4 (positive), the phase shift is 0 (no horizontal shift), and the vertical shift is –2. Thus the setup and several conclusions are incorrect.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the phase shift as 1/2 to the right. The function y = 4 cos(pi x) - 2 has a phase shift of 0 (or any integer multiple of the period). The step claiming h = 1/2 likely stems from confusing the phase shift with the x-intercept or the location of a specific feature, but the standard form y = A cos(Bx - C) + D yields C=0, so h=0. Additionally, converting to sine form in step 1 is unnecessary and confusingly executed.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution incorrectly identifies the phase shift as 1/2 to the right. The function y = 4 cos(pi x) - 2 has a phase shift of 0 (or any integer multiple of the period). The step claiming h = 1/2 likely stems from confusing the phase shift with the x-intercept or the location of a specific feature, but the standard form y = A cos(Bx - C) + D yields C=0, so h=0. Additionally, converting to sine form in step 1 is unnecessary and confusingly executed.gpt-oss:20b: fail (error) 2026-10-20 — The solution incorrectly rewrites the function as A·sin(B(x−h))+D with A = –4 and h = ½. The amplitude of 4 cos(πx)–2 is 4 (positive), the phase shift is 0 (no horizontal shift), and the vertical shift is –2. Thus the setup and several conclusions are incorrect.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 = 4 cos(pi x) - 2 has a phase shift of 0. While it is true that cos(pi x) = -sin(pi(x - 1/2)), converting to a negative sine function changes the phase reference but does not imply the original cosine function has a non-zero phase shift in its standard form. The stated answer claims a phase shift of 1/2, which is incorrect for the given cosine function.gpt-oss:20b: fail (error) 2026-10-19 — The solution incorrectly rewrites the function as A·sin(B(x−h))+D with A = −4 and h = ½. The given function is 4 cos(πx)−2, so the amplitude is +4, the period is 2, the phase shift is 0 (no shift), and the vertical shift is −2. The sign of A and the value of h are wrong, which misleads the student about the correct parameters.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 = 4 cos(pi x) - 2 has a phase shift of 0. The step claiming h = 1/2 is derived from an incorrect conversion or application of the phase shift formula for cosine, and the final answer contradicts the standard form of the given function.gpt-oss:20b: fail (error) 2026-10-19 — The first sentence incorrectly identifies A as –4 and h as ½, implying a phase shift of ½ to the right. In fact, y=4 cos(πx)–2 already has A=4, B=π, h=0, D=–2, so there is no phase shift. This misidentification leads to incorrect conclusions about amplitude and phase shift.
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 2026-10-20 with SymPy 1.14.0.