Math Problem Statement
Solution
We are asked to find a trigonometric function of the form or that matches the graph shown.
Steps to determine the equation:
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Amplitude : The amplitude is the distance from the centerline (vertical shift) to the peak. From the graph, the maximum value is and the minimum value is , so the amplitude is:
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Period and : The period is the distance between two consecutive peaks or troughs. Observing the graph, the peaks are at and , meaning the period is: The relationship between the period and is given by: So, solving for :
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Phase Shift: The graph resembles a cosine wave, as it starts at a maximum at . Since the general cosine function has its peak at , this graph is shifted to the left by 9 units. Therefore, the phase shift is .
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Vertical Shift : The graph oscillates symmetrically about , so there is no vertical shift. Thus, .
Final Equation:
The equation that matches the graph is:
Let me know if you need further details or explanations!
Here are 5 related questions you can explore:
- How do we calculate the amplitude for a trigonometric function?
- What is the general formula for the period of sine and cosine functions?
- How does the phase shift affect the graph of a trigonometric function?
- Can this graph be represented using a sine function instead of a cosine function?
- What role does the vertical shift play in modifying a trigonometric function?
Tip: The cosine function peaks at , while the sine function starts from 0, which helps decide which form to use when modeling real-life periodic graphs.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Graph Analysis
Amplitude
Periodicity
Phase Shift
Vertical Shifts
Formulas
y = A sin(kx) + C
y = A cos(kx) + C
Amplitude = (max - min) / 2
Period T = 2π / k
Vertical Shift = C
Theorems
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Suitable Grade Level
Grades 9-12
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