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Amplitude, period and phase shift

Problem 13.394 · medium

For \( \displaystyle y = 2 \sin{\left(x + \frac{\pi}{3} \right)} - 2 \), find the amplitude, the period, the phase shift (with its direction) and the vertical shift.
  1. Write it as A·sin(B(x − h)) + D: A = 2, B = 1, h = -pi/3, D = -2.
  2. \[ \frac{2 \pi}{1} = 2 \pi \]
    Period = 2π/B.✓ Proved
  3. \[ \left(2 - 2 \sin{\left(x + \frac{\pi}{3} \right)}\right) + \left. 2 \sin{\left(x + \frac{\pi}{3} \right)} - 2 \right|_{\substack{ x=x + 2 \pi }} = 0 \]
    Shifting x by one period gives the same function.✓ Proved
  4. \[ \frac{\left(-1\right) \pi}{1 \cdot 3} = - \frac{\pi}{3} \]
    Phase shift h = C/B.✓ Proved
  5. \[ \left. 2 \sin{\left(x + \frac{\pi}{3} \right)} - 2 \right|_{\substack{ x=\frac{\pi}{6} }} = 0 \]
    At x = pi/6 the curve is 2 from its midline y = -2: that is the amplitude.✓ Proved
Answer \( \text{amplitude } 2,\ \text{period } 2 \pi,\ \text{phase shift } \frac{\pi}{3}\text{ to the left},\ \text{vertical shift } -2 \)

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
LineStatusChecked byDetail
1Not checked—a sentence; read, not computed
2✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
3✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
4✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
5✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
answer, a second way✓ Checked independentlysympy 1.14.0 + mpmath 1.3.0max 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: pass
  • qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution incorrectly identifies the phase shift as -pi/3 (or pi/3 to the left) in line 1 and line 4. For y = 2 sin(x + pi/3) - 2, the phase shift is -pi/3, which corresponds to a shift of pi/3 to the left. However, line 1 states h = -pi/3, and line 4 calculates h = -pi/3. The stated answer says 'pi/3 to the left', which is consistent with h = -pi/3. Wait, let's re-read carefully. Standard form: y = A sin(B(x - h)) + D. Given: y = 2 sin(1(x - (-pi/3))) - 2. So h = -pi/3. A phase shift of h means the graph is shifted by h. If h is negative, it shifts left. So shift is pi/3 to the left. Let's look at line 1: 'h = -pi/3'. This is correct for the form B(x-h). Line 4: 'Phase shift h = C/B'. Here C is the coefficient inside the sine if written as sin(Bx + C). y = 2 sin(x + pi/3) - 2. B = 1, C = pi/3. Phase shift is usually defined as -C/B. -C/B = -(pi/3)/1 = -pi/3. So the shift is -pi/3. A shift of -pi/3 is pi/3 to the left. The stated answer says: 'phase shift pi/3 to the left'. This is correct. Let's check line 5. 'At x = pi/6 the curve is 2 from its midline y = -2: that is the amplitude.' Midline is y = -2. Amplitude is 2. So we are looking for a point where y = -2 + 2 = 0 or y = -2 - 2 = -4. Let's check x = pi/6. y(pi/6) = 2 sin(pi/6 + pi/3) - 2 = 2 sin(pi/2) - 2 = 2(1) - 2 = 0. Distance from midline (-2) to 0 is |0 - (-2)| = 2. This confirms the amplitude is 2. Is there an error? Line 1 says 'h = -pi/3'. Line 4 says 'Phase shift h = C/B'. If the formula used is h = C/B, then h = (pi/3)/1 = pi/3. But line 4 calculates '-pi/3/1 = -pi/3'. This implies the formula used was -C/B, but the text says 'h = C/B'. This is a contradiction in the sentence description vs the calculation. If the text says 'h = C/B', it should calculate pi/3. If it calculates -pi/3, it should say 'h = -C/B'. Furthermore, if h is defined as the value in (x-h), then h = -pi/3. If the text claims 'Phase shift h = C/B', it is stating a false formula for the standard form y = A sin(B(x-h)) + D where the argument is Bx+C. In y = A sin(Bx+C), the phase shift is -C/B. The sentence says 'Phase shift h = C/B'. This is mathematically incorrect for the standard definition of phase shift relative to the origin for sin(Bx+C). Also, line 1 says h = -pi/3. Line 4 calculates -pi/3. But the label 'h = C/B' is wrong because C/B would be pi/3. Therefore, the sentence in line 4 contains a false statement about the formula for phase shift.
Every verdict on record (6)
  • qwen3.6:27b-mlx: fail (style) 2026-10-20 — [domain objection, downgraded to style] The solution incorrectly identifies the phase shift as -pi/3 (or pi/3 to the left) in line 1 and line 4. For y = 2 sin(x + pi/3) - 2, the phase shift is -pi/3, which corresponds to a shift of pi/3 to the left. However, line 1 states h = -pi/3, and line 4 calculates h = -pi/3. The stated answer says 'pi/3 to the left', which is consistent with h = -pi/3. Wait, let's re-read carefully. Standard form: y = A sin(B(x - h)) + D. Given: y = 2 sin(1(x - (-pi/3))) - 2. So h = -pi/3. A phase shift of h means the graph is shifted by h. If h is negative, it shifts left. So shift is pi/3 to the left. Let's look at line 1: 'h = -pi/3'. This is correct for the form B(x-h). Line 4: 'Phase shift h = C/B'. Here C is the coefficient inside the sine if written as sin(Bx + C). y = 2 sin(x + pi/3) - 2. B = 1, C = pi/3. Phase shift is usually defined as -C/B. -C/B = -(pi/3)/1 = -pi/3. So the shift is -pi/3. A shift of -pi/3 is pi/3 to the left. The stated answer says: 'phase shift pi/3 to the left'. This is correct. Let's check line 5. 'At x = pi/6 the curve is 2 from its midline y = -2: that is the amplitude.' Midline is y = -2. Amplitude is 2. So we are looking for a point where y = -2 + 2 = 0 or y = -2 - 2 = -4. Let's check x = pi/6. y(pi/6) = 2 sin(pi/6 + pi/3) - 2 = 2 sin(pi/2) - 2 = 2(1) - 2 = 0. Distance from midline (-2) to 0 is |0 - (-2)| = 2. This confirms the amplitude is 2. Is there an error? Line 1 says 'h = -pi/3'. Line 4 says 'Phase shift h = C/B'. If the formula used is h = C/B, then h = (pi/3)/1 = pi/3. But line 4 calculates '-pi/3/1 = -pi/3'. This implies the formula used was -C/B, but the text says 'h = C/B'. This is a contradiction in the sentence description vs the calculation. If the text says 'h = C/B', it should calculate pi/3. If it calculates -pi/3, it should say 'h = -C/B'. Furthermore, if h is defined as the value in (x-h), then h = -pi/3. If the text claims 'Phase shift h = C/B', it is stating a false formula for the standard form y = A sin(B(x-h)) + D where the argument is Bx+C. In y = A sin(Bx+C), the phase shift is -C/B. The sentence says 'Phase shift h = C/B'. This is mathematically incorrect for the standard definition of phase shift relative to the origin for sin(Bx+C). Also, line 1 says h = -pi/3. Line 4 calculates -pi/3. But the label 'h = C/B' is wrong because C/B would be pi/3. Therefore, the sentence in line 4 contains a false statement about the formula for phase shift.
  • gpt-oss:20b: pass 2026-10-20
  • qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 1 incorrectly identifies h as -pi/3. For the form A*sin(B(x-h)), expanding gives Bx - Bh. Matching Bx - Bh to x + pi/3 (where B=1) yields -h = pi/3, so h = -pi/3. However, the standard definition of phase shift is usually given as -C/B for A*sin(Bx+C). Here C=pi/3, B=1, so shift is -pi/3 (left). Step 1 says h=-pi/3, which implies the form is sin(1(x - (-pi/3))) = sin(x + pi/3). This is algebraically correct. But Step 4 calculates h = C/B = pi/3/1 = pi/3? No, Step 4 says -pi/3/1 = -pi/3. Wait, Step 1 says h = -pi/3. Step 4 says Phase shift h = C/B. If the form is A*sin(B(x-h)), then h is the shift. If the form is A*sin(Bx+C), the shift is -C/B. The solution mixes conventions or defines h inconsistently. Step 1 says h=-pi/3. Step 4 calculates -pi/3/1. This looks like it's calculating -C/B. But it labels it 'h'. If h is the shift, h = -pi/3. The final answer says 'pi/3 to the left', which is equivalent to a shift of -pi/3. The error is in Step 1's labeling or Step 4's justification. Step 1 says 'Write it as ... h = -pi/3'. This is correct for the form sin(B(x-h)). Step 4 says 'Phase shift h = C/B'. This formula is wrong for the form sin(B(x-h)). For sin(B(x-h)), h is the shift. For sin(Bx+C), shift is -C/B. The solution uses the formula for the second form but claims it equals h from the first form, creating a contradiction in notation/logic. Specifically, if h=-pi/3, then h != C/B (which is pi/3). It should be h = -C/B. The text says 'h = C/B' then computes '-pi/3/1'. It seems to mean '-C/B' but writes 'C/B'. This is a misleading/incorrect statement of the formula.
  • gpt-oss:20b: fail (style) 2026-10-19 — [domain objection, downgraded to style] The solution contains several incorrect or unclear statements: the phase‑shift calculation is not justified (C is undefined), the period check uses a nonsensical equation, and the amplitude test is wrong. The setup is correct, but the explanatory sentences are mathematically flawed and would mislead a student.
  • qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution incorrectly identifies the phase shift as -pi/3 (left) when the form sin(x + pi/3) implies a shift of -pi/3, which is pi/3 to the left. Wait, let's re-read carefully. The standard form is sin(B(x-h)). Here we have sin(1(x - (-pi/3))). So h = -pi/3. A shift of h means the graph is shifted by h. So the shift is -pi/3, which is pi/3 to the left. The stated answer says 'pi/3 to the left'. The solution says h = -pi/3. This is consistent. Let's look at step 5. It claims that at x=pi/6, the curve is 2 from its midline. Midline is y=-2. Value is 2*sin(pi/6 + pi/3) - 2 = 2*sin(pi/2) - 2 = 2(1) - 2 = 0. Distance from midline -2 is |0 - (-2)| = 2. This is correct. However, the definition of phase shift in step 1 is slightly ambiguous in notation but the value h=-pi/3 is correct for the form A sin(B(x-h)). The stated answer says 'phase shift pi/3 to the left'. This corresponds to h = -pi/3. The solution calculates h = -pi/3. This seems correct. Why would it fail? Let's check the period calculation. Period = 2pi/B = 2pi/1 = 2pi. Correct. Amplitude = |A| = 2. Correct. Vertical shift D = -2. Correct. Is there a subtle error? Step 1 says 'Write it as A·sin(B(x − h)) + D: A = 2, B = 1, h = -pi/3, D = -2.' The original function is 2 sin(x + pi/3) - 2. This is 2 sin(1(x - (-pi/3))) - 2. So h = -pi/3. The phase shift is h. So the shift is -pi/3. This is pi/3 to the left. The stated answer matches. Step 4 calculates -pi/3/1 = -pi/3. This is consistent. Step 5 checks the amplitude. It picks a specific point x=pi/6 where the sine term is 1. This is a valid check for amplitude. Is there anything wrong? The prompt asks to find the phase shift with its direction. The solution provides h=-pi/3. It does not explicitly state 'pi/3 to the left' in the text, but the stated answer does. The review is of the solution lines. Line 1 identifies h=-pi/3. This is the mathematical value of the shift. Usually, 'phase shift' refers to the value h. The direction is implied by the sign. The stated answer interprets this correctly. The solution seems sound. Wait, look at step 3. It checks periodicity. It's a bit weirdly formatted but algebraically correct. Look at step 5 again. 'At x = pi/6 the curve is 2 from its midline y = -2: that is the amplitude.' This is a valid way to verify the amplitude. Is there a convention issue? Some texts define phase shift as -C/B for sin(Bx+C). Here C=pi/3, B=1, so -C/B = -pi/3. This matches h. Everything seems correct. Let me look closer at the 'adversarial' nature. Is there a trap? 'Phase shift ... with its direction'. The solution gives h=-pi/3. It doesn't explicitly say 'left'. But the stated answer does. The solution lines are the derivation. The derivation is correct. Maybe the issue is that step 1 defines h as -pi/3, but step 4 calculates -pi/3/1. Where did the negative sign come from in step 4? Step 4 says 'Phase shift h = C/B'. If the form is sin(Bx+C), then phase shift is -C/B. If the form is sin(B(x-h)), then h is the shift. In step 1, the form is A sin(B(x-h)). So h is the shift. In step 4, it calculates -pi/3/1. This implies it used the formula -C/B where C=pi/3. But step 1 said h=-pi/3. If h=-pi/3, then the shift is -pi/3. The calculation in step 4 is consistent with the value in step 1. However, the label 'h = C/B' in step 4 is technically incorrect if C is the coefficient inside the sine in the form sin(Bx+C). Usually, phase shift = -C/B. If the solution writer thinks h = C/B, they might have made a sign error in their head, but got the right number because they plugged in -pi/3? No, step 4 calculates -pi/3/1. The numerator is -pi/3. Where did -pi/3 come from? It came from the term (x + pi/3). If one writes x + pi/3 = x - (-pi/3), then h = -pi/3. If one uses the formula -C/B, C=pi/3, so -pi/3/1 = -pi/3. The text in step 4 says 'h = C/B'. This is a false statement about the formula. The formula is h = -C/B (for sin(Bx+C)) or h is just the value inside the parenthesis with sign flipped. Stating 'h = C/B' is misleading/wrong because it ignores the negative sign required to convert from the 'C' form to the 'h' form, or it mislabels the variable. If the student learns 'phase shift is C/B', they will get the wrong sign for sin(x + pi/3) -> shift pi/3 (right) instead of left. This is a significant conceptual error in the explanation.
  • gpt-oss:20b: pass 2026-10-19

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.

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