Generalization, model comparison, proof, and technology-assisted validation.
Directions: Show exact work before decimals, label domains or intervals, and explain calculator use.
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For y=4sin(3x)−2, state the amplitude.
2. {'id': 'u05-l07', 'number': 7, 'title': 'Sinusoidal Regression and Periodic Data', 'family': 'triggraph', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
For y=4sin(3x)−2, state the period.
3. {'id': 'u05-l07', 'number': 7, 'title': 'Sinusoidal Regression and Periodic Data', 'family': 'triggraph', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
For y=2cos(x−π/4)+1, state the phase shift.
4. {'id': 'u05-l07', 'number': 7, 'title': 'Sinusoidal Regression and Periodic Data', 'family': 'triggraph', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
State the midline of y=−3sin(2x)+5.
5. {'id': 'u05-l07', 'number': 7, 'title': 'Sinusoidal Regression and Periodic Data', 'family': 'triggraph', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
State one vertical asymptote of y=tan x.
6. {'id': 'u05-l07', 'number': 7, 'title': 'Sinusoidal Regression and Periodic Data', 'family': 'triggraph', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
A periodic quantity ranges from 8 to 20. Find amplitude and midline.
7. {'id': 'u05-l08', 'number': 8, 'title': 'Harmonic Comparisons and Model Limitations', 'family': 'limits', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
Find the average rate of change of x² from 2 to 5.
8. {'id': 'u05-l08', 'number': 8, 'title': 'Harmonic Comparisons and Model Limitations', 'family': 'limits', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
Simplify [(x+h)²−x²]/h.
9. {'id': 'u05-l08', 'number': 8, 'title': 'Harmonic Comparisons and Model Limitations', 'family': 'limits', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
Evaluate lim_{x→3}(2x+1).
10. {'id': 'u05-l08', 'number': 8, 'title': 'Harmonic Comparisons and Model Limitations', 'family': 'limits', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
Evaluate lim_{x→2}(x²−4)/(x−2).
11. {'id': 'u05-l08', 'number': 8, 'title': 'Harmonic Comparisons and Model Limitations', 'family': 'limits', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
Can a two-sided limit exist if one-sided limits differ?
12. {'id': 'u05-l08', 'number': 8, 'title': 'Harmonic Comparisons and Model Limitations', 'family': 'limits', 'week': 15, 'unit': 5, 'unit_title': 'Trigonometric Functions, Graphs & Periodic Models', 'essential': 'How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?', 'project': 'Periodic Phenomenon Model & Residual Analysis'}
Name the discontinuity in (x²−1)/(x−1) at x=1.