write the equation of the following sine curves.

15.

16.

Question

write the equation of the following sine curves.

15.

16.

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Trigonometric Functions", "Sine Wave Equations", "Amplitude and Period" ], "current_concepts": [ "Trigonometric Functions", "Sine Wave Equations", "Amplitude and Period" ] } ## Analyze the graph for Question 15 $$ \begin{aligned} &\text{Amplitude } (A): \text{Maximum value is } 2, \text{ minimum value is } -2 \implies A = 2\ &\text{Period } (T): \text{The wave completes one full cycle from } 0 \text{ to } \pi \implies T = \pi\ &\text{Frequency coefficient } (b): b = \frac{2\pi}{T} = \frac{2\pi}{\pi} = 2\ &\text{Phase shift and vertical shift}: \text{Starts at } (0,0) \text{ going upwards} \implies \text{No shifts}\ &\text{Equation}: y = 2\sin(2x) \end{aligned} $$ ## Analyze the graph for Question 16 $$ \begin{aligned} &\text{Amplitude } (A): \text{Maximum value is } 3, \text{ minimum value is } -3 \implies A = 3\ &\text{Period } (T): \text{The wave completes one full cycle from } 0 \text{ to } 2\pi \implies T = 2\pi\ &\text{Frequency coefficient } (b): b = \frac{2\pi}{T} = \frac{2\pi}{2\pi} = 1\ &\text{Phase shift and vertical shift}: \text{Starts at } (0,0) \text{ going downwards} \implies \text{Reflected across the x-axis}\ &\text{Equation}: y = -3\sin(x) \end{aligned} $$ ## Question 15 $$y = 2\sin(2x)$$ ## Question 16 $$y = -3\sin(x)$$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Functions" ] }

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Question

Explanation:

Analyze the graph for Question 15

Analyze the graph for Question 16

Answer:

Question 15

Question 16