Question
Write the equation of the periodic function based on the graph. (1 point) Responses y=4cosx y equals 4 cosine x y=−4cosx y equals negative 4 cosine x y=−14cosx y equals negative Start Fraction 1 over 4 End Fraction cosine x y=−4sinx
Answers
GPT-4o mini
To determine the equation of a periodic function based on its graph, you need to consider the amplitude, the vertical shift, and whether it's a sine or cosine function.
1. **Amplitude**: The amplitude is the maximum value of the function. If the function oscillates from -A to A, then the amplitude is A. For example, if you see that the graph peaks at 4 and troughs at -4, the amplitude is 4.
2. **Cosine vs. Sine**: If the graph starts at its maximum point, it likely represents a cosine function (as cosine starts at the maximum). If it starts at the midline and goes up first, it is likely a sine function.
3. **Vertical Shift**: If the graph is centered around a line that is not y=0, this means there is a vertical shift.
4. **Period**: This is determined by how quickly the function repeats. For cosine and sine, this is typically influenced by the coefficient of x in the function.
Given the options provided, and without additional details on shifts or periodicity from the graph, if we assume maximum peaks of 4 and troughs of -4, and the function is cosine:
- The amplitude is 4.
- Assuming there is no vertical shift and it starts at the maximum, the equation would be:
**y = 4 cos x** (if starting at maximum).
If, however, it's reflected or if further details change it, you might choose one of the other options. But based on the peak-to-trough description, the most fitting choice is:
**y = 4 cos x**.
1. **Amplitude**: The amplitude is the maximum value of the function. If the function oscillates from -A to A, then the amplitude is A. For example, if you see that the graph peaks at 4 and troughs at -4, the amplitude is 4.
2. **Cosine vs. Sine**: If the graph starts at its maximum point, it likely represents a cosine function (as cosine starts at the maximum). If it starts at the midline and goes up first, it is likely a sine function.
3. **Vertical Shift**: If the graph is centered around a line that is not y=0, this means there is a vertical shift.
4. **Period**: This is determined by how quickly the function repeats. For cosine and sine, this is typically influenced by the coefficient of x in the function.
Given the options provided, and without additional details on shifts or periodicity from the graph, if we assume maximum peaks of 4 and troughs of -4, and the function is cosine:
- The amplitude is 4.
- Assuming there is no vertical shift and it starts at the maximum, the equation would be:
**y = 4 cos x** (if starting at maximum).
If, however, it's reflected or if further details change it, you might choose one of the other options. But based on the peak-to-trough description, the most fitting choice is:
**y = 4 cos x**.