Amplitude, period and phase shift
Problem 13.272 · easy
For \( \displaystyle y = 4 \cos{\left(2 x \right)} - 3 \), 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 = 4, B = 2, h = 0, D = -3.
- \[ \frac{2 \pi}{2} = \pi \]Period = 2π/B.✓ Proved
- \[ \left(3 - 4 \cos{\left(2 x \right)}\right) + \left. 4 \cos{\left(2 x \right)} - 3 \right|_{\substack{ x=x + \pi }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ \frac{0}{2} = 0 \]Phase shift h = C/B.✓ Proved
- \[ \left. 4 \cos{\left(2 x \right)} - 3 \right|_{\substack{ x=0 }} = 1 \]At x = 0 the curve is 4 from its midline y = -3: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 4,\ \text{period } \pi,\ \text{phase shift } 0,\ \text{vertical shift } -3 \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in 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 third sentence attempts to show that shifting by one period gives the same function, but the equation written is incorrect (it includes an unnecessary negative sign and does not properly demonstrate the periodicity). This could mislead a student into thinking the algebraic manipulation is valid when it is not.qwen3.6:27b-mlx: fail (misleading) — Step 5 incorrectly defines amplitude as the value of the function at x=0 relative to the midline, rather than the coefficient A. While the numerical result is correct for this specific function, the reasoning is false (amplitude is not generally f(0) - D) and would teach a student an incorrect definition.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — Step 5 incorrectly defines amplitude as the value of the function at x=0 relative to the midline, rather than the coefficient A. While the numerical result is correct for this specific function, the reasoning is false (amplitude is not generally f(0) - D) and would teach a student an incorrect definition.gpt-oss:20b: fail (misleading) 2026-10-18 — The third sentence attempts to show that shifting by one period gives the same function, but the equation written is incorrect (it includes an unnecessary negative sign and does not properly demonstrate the periodicity). This could mislead a student into thinking the algebraic manipulation is valid when it is not.qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] Line 5 claims that the value of the function at x=0 being 1 proves the amplitude is 4. This is logically invalid; the amplitude is defined as |A|, not derived from a single point's distance from the midline without establishing that this point is a maximum or minimum. While the distance happens to be the amplitude here, the reasoning presented is flawed and misleading.gpt-oss:20b: fail (style) 2026-10-16 — [domain objection, downgraded to style] Sentences 3 and 4 are confusing and incorrectly justify the phase shift. Sentence 3 uses an unnecessary negative sign and a nonsensical equation; sentence 4 refers to an undefined variable C. The conclusion that the phase shift is 0 is correct, but the reasoning is misleading and would not teach a student the proper method for determining phase shift.qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 5 incorrectly defines amplitude as the function value at x=0 (which is 1), rather than the coefficient A (which is 4). This is a fundamental conceptual error.gpt-oss:20b: fail (error) 2026-10-15 — The solution incorrectly calculates the amplitude: it evaluates the function at x=0 and concludes the amplitude is 1, whereas the amplitude is the absolute value of A, which is 4. Additionally, the justification for the phase shift is unclear and the equation in step 3 is nonsensical.
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.