Amplitude, period and phase shift
Problem 13.97 · medium
For \( \displaystyle y = 2 \cos{\left(4 x + 8 \right)} \), 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 = 2, B = 4, h = -2, D = 0.
- \[ \frac{2 \pi}{4} = \frac{\pi}{2} \]Period = 2π/B.✓ Proved
- \[ - 2 \cos{\left(4 x + 8 \right)} + \left. 2 \cos{\left(4 x + 8 \right)} \right|_{\substack{ x=x + \frac{\pi}{2} }} = 0 \]Shifting x by one period gives the same function.✓ Proved
- \[ - \frac{8}{4} = -2 \]Phase shift h = C/B.✓ Proved
- \[ \left. 2 \cos{\left(4 x + 8 \right)} \right|_{\substack{ x=-2 }} = 2 \]At x = -2 the curve is 2 from its midline y = 0: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 2,\ \text{period } \frac{\pi}{2},\ \text{phase shift } 2\text{ to the left},\ \text{vertical shift } 0 \)
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 (error) — Sentence 3 incorrectly asserts that shifting by one period yields the same function using a nonsensical expression, and sentence 5 misidentifies the amplitude by evaluating the function at x = –2 instead of using the absolute value of A. The setup is correct, but these sentences contain false or misleading statements.qwen3.6:27b-mlx: fail (misleading) — The solution incorrectly identifies the phase shift as 2 to the left (h = -2) instead of 2 to the right (h = 2). The standard form y = A cos(B(x - h)) + D requires factoring out B: 2 cos(4(x + 2)), so h = -2, which corresponds to a shift of -2 (2 units left) is actually correct? Wait. Let's re-evaluate. y = 2 cos(4x + 8) = 2 cos(4(x + 2)). The form is A cos(B(x - h)). So x - h = x + 2 => -h = 2 => h = -2. A shift of h = -2 means the graph is shifted 2 units to the left. The stated answer says '2 to the left'. The solution says 'h = -2'. This is consistent. Why did I think it was wrong? Let's check the period. Period = 2pi/4 = pi/2. Correct. Amplitude = 2. Correct. Vertical shift = 0. Correct. Phase shift: 4x + 8 = 0 => x = -2. So the 'start' of the cosine cycle is at x = -2. This is 2 units to the left of the origin. The solution is actually correct. Let me look closer at the prompt's request. 'find the amplitude, the period, the phase shift (with its direction) and the vertical shift.' The stated answer is 'phase shift 2 to the left'. The solution derives h = -2. In the form A cos(B(x-h)), h is the phase shift. If h = -2, the shift is -2, i.e., 2 to the left. This seems correct. Is there a subtle error? Step 1 says 'Write it as A·cos(B(x − h)) + D: A = 2, B = 4, h = -2, D = 0.' This implies 2 cos(4(x - (-2))) = 2 cos(4(x+2)) = 2 cos(4x+8). This is algebraically correct. Step 4 calculates -8/4 = -2. This is the standard formula for phase shift C/B where y = A cos(Bx + C). Here C=8, B=4, so shift is -C/B = -2. This is correct. Step 5 checks the value at x=-2. cos(4(-2)+8) = cos(0) = 1. 2*1 = 2. This is the maximum. The midline is 0. So it is 2 units from midline. This confirms amplitude. Everything seems correct. Why would it fail? Maybe the definition of phase shift? Some texts define phase shift as just the value h, others as the magnitude and direction. The answer provides direction. The solution provides h=-2. This is consistent. Is there a 'misleading' aspect? Perhaps the notation 'h = -2' is ambiguous without stating 'shift is -2'? No, that's standard. Let's look at Step 3. It checks periodicity. It's irrelevant to finding the parameters but not wrong. Let's look at Step 1 again. 'Write it as... A=2, B=4, h=-2, D=0'. This is a statement of identification. It is correct. I will mark it as pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution incorrectly identifies the phase shift as 2 to the left (h = -2) instead of 2 to the right (h = 2). The standard form y = A cos(B(x - h)) + D requires factoring out B: 2 cos(4(x + 2)), so h = -2, which corresponds to a shift of -2 (2 units left) is actually correct? Wait. Let's re-evaluate. y = 2 cos(4x + 8) = 2 cos(4(x + 2)). The form is A cos(B(x - h)). So x - h = x + 2 => -h = 2 => h = -2. A shift of h = -2 means the graph is shifted 2 units to the left. The stated answer says '2 to the left'. The solution says 'h = -2'. This is consistent. Why did I think it was wrong? Let's check the period. Period = 2pi/4 = pi/2. Correct. Amplitude = 2. Correct. Vertical shift = 0. Correct. Phase shift: 4x + 8 = 0 => x = -2. So the 'start' of the cosine cycle is at x = -2. This is 2 units to the left of the origin. The solution is actually correct. Let me look closer at the prompt's request. 'find the amplitude, the period, the phase shift (with its direction) and the vertical shift.' The stated answer is 'phase shift 2 to the left'. The solution derives h = -2. In the form A cos(B(x-h)), h is the phase shift. If h = -2, the shift is -2, i.e., 2 to the left. This seems correct. Is there a subtle error? Step 1 says 'Write it as A·cos(B(x − h)) + D: A = 2, B = 4, h = -2, D = 0.' This implies 2 cos(4(x - (-2))) = 2 cos(4(x+2)) = 2 cos(4x+8). This is algebraically correct. Step 4 calculates -8/4 = -2. This is the standard formula for phase shift C/B where y = A cos(Bx + C). Here C=8, B=4, so shift is -C/B = -2. This is correct. Step 5 checks the value at x=-2. cos(4(-2)+8) = cos(0) = 1. 2*1 = 2. This is the maximum. The midline is 0. So it is 2 units from midline. This confirms amplitude. Everything seems correct. Why would it fail? Maybe the definition of phase shift? Some texts define phase shift as just the value h, others as the magnitude and direction. The answer provides direction. The solution provides h=-2. This is consistent. Is there a 'misleading' aspect? Perhaps the notation 'h = -2' is ambiguous without stating 'shift is -2'? No, that's standard. Let's look at Step 3. It checks periodicity. It's irrelevant to finding the parameters but not wrong. Let's look at Step 1 again. 'Write it as... A=2, B=4, h=-2, D=0'. This is a statement of identification. It is correct. I will mark it as pass.gpt-oss:20b: fail (error) 2026-10-17 — Sentence 3 incorrectly asserts that shifting by one period yields the same function using a nonsensical expression, and sentence 5 misidentifies the amplitude by evaluating the function at x = –2 instead of using the absolute value of A. The setup is correct, but these sentences contain false or misleading statements.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the parameters A, B, h, and D by rewriting the function in standard form. The derived amplitude, period, phase shift, and vertical shift match the stated answer and are mathematically sound.gpt-oss:20b: fail (error) 2026-10-14 — Equation 3 is incorrect: it mixes a negative cosine and an unrelated substitution, yet claims that shifting by one period yields the same function. The algebraic statement does not follow from the given function and would mislead a student.qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution incorrectly identifies the phase shift as 2 to the left. The phase shift is -2, which corresponds to a shift of 2 units to the left, but the standard form y = A cos(B(x - h)) + D implies h = -2, so the shift is indeed 2 to the left. Wait, let me re-read carefully. The problem asks for phase shift with direction. h = -2 means shift is -2 (left). The stated answer says '2 to the left'. This is correct. Let me check the derivation. Step 1 says h = -2. Step 4 calculates -8/4 = -2. This is correct. Why did I think it was wrong? Ah, let's look at Step 5. It claims that at x = -2, the curve is 2 from its midline, which is the amplitude. While true that the value is 2, this is not the definition of amplitude. Amplitude is |A|. The fact that it reaches 2 at the phase shift is a consequence, not the definition. However, the bigger issue is usually how phase shift is defined. Some texts define phase shift as -C/B. Here C=8, B=4, so -8/4 = -2. A shift of -2 is 2 units left. The stated answer is correct. Is there an error in the sentences? Step 1: 'Write it as A·cos(B(x − h)) + D: A = 2, B = 4, h = -2, D = 0.' This is correct. 2cos(4(x - (-2))) = 2cos(4x + 8). Step 2: Period calculation is correct. Step 3: Checks periodicity. Correct. Step 4: Calculates h. Correct. Step 5: 'At x = -2 the curve is 2 from its midline y = 0: that is the amplitude.' This sentence is misleading/wrong in its logic. It implies that the amplitude is defined by the value at the phase shift. While the value at the phase shift is indeed the amplitude (for a cosine function starting at a peak), this is not the general definition. More importantly, is the phase shift direction correct? h = -2. The shift is h. So the shift is -2. '2 to the left' is the correct interpretation. Let's look closer at Step 1. It says 'h = -2'. In the form cos(B(x-h)), if h=-2, then (x - (-2)) = (x+2). So the argument is 4(x+2) = 4x+8. This matches. The phase shift is h. So the shift is -2. A shift of -2 is 2 units to the left. The stated answer is '2 to the left'. This is consistent. Is there a subtle error? Let's check the sign convention. Some sources define phase shift as -C/B. Others define it as the value h in B(x-h). Both yield -2. Let's re-read the prompt's instruction: 'find the amplitude, the period, the phase shift (with its direction) and the vertical shift.' Stated answer: amplitude 2, period pi/2, phase shift 2 to the left, vertical shift 0. Solution Step 1: h = -2. Solution Step 4: -8/4 = -2. If h = -2, the shift is 2 units to the left. This is correct. Why might this be a fail? Maybe the sentence in Step 5 is the problem. 'At x = -2 the curve is 2 from its midline y = 0: that is the amplitude.' This is a non-sequitur or a weak justification. The amplitude is defined as |A|, which is 2. The fact that f(-2) = 2 confirms A=2 (since cos(0)=1), but stating 'that is the amplitude' suggests the value *is* the definition, or that one must evaluate at the phase shift to find amplitude. It's a bit sloppy but not necessarily 'wrong' in a way that teaches false math, unless it implies amplitude is always f(h). For sine, f(h) would be 0. So this statement is false for sine functions. It teaches a student that amplitude is the value of the function at the phase shift, which is only true for cosine (and -cosine) forms, not sine. This is misleading. Also, Step 1 says 'Write it as... h = -2'. It doesn't explicitly state the direction 'left' in the sentence, but the final answer does. The question asks to review the solution. The solution's sentences are the focus. Step 5 is definitely misleading because it generalizes a property of cosine to the definition of amplitude.gpt-oss:20b: fail (error) 2026-10-14 — Sentence 3 is nonsensical and does not correctly justify that shifting by one period leaves the function unchanged; the algebraic expression is incorrect. Sentence 4 incorrectly states the formula for the phase shift as h = C/B (missing the negative sign) before computing h = -C/B. These errors misrepresent the reasoning and could mislead a student.
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.