Math Problem Statement
Solution
The question asks for the graph of .
Key properties of the function :
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Amplitude: The coefficient of the sine function is 2, meaning the amplitude is 2. This indicates that the graph will oscillate between -2 and 2 on the y-axis.
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Period: The period of a sine function is given by , where is the coefficient of inside the sine. In this case, , so the period is: This means the sine wave will complete one full cycle every units along the x-axis.
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Frequency: Since the sine function repeats every , it will have a higher frequency than the regular sine function.
Identifying the correct graph:
- The amplitude is 2, so the graph should have peaks at 2 and valleys at -2.
- The period is , so the graph should complete one cycle in that span.
The second graph from the left matches these properties, as it has the correct amplitude of 2 and period of .
Answer: The correct graph is the second one from the left.
Would you like more details on any of these points?
Here are five related questions you might find helpful:
- How do you determine the amplitude of a sine function from its equation?
- What is the relationship between the coefficient of inside the sine function and the period?
- How do transformations (like vertical shifts) affect the graph of a sine function?
- What would the graph look like if the equation was ?
- How does the frequency of a sine wave change when you change the coefficient of ?
Tip:
When identifying graphs of sine or cosine functions, always start by analyzing the amplitude and period, as these will give you the key characteristics to look for.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Amplitude
Period
Sine Functions
Formulas
Amplitude = 2
Period = 2π/3
Theorems
Properties of sine functions
Transformation of trigonometric functions
Suitable Grade Level
Grades 10-12
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