Question
Simultaneous Transformations of Periodic Functions Practice
Complete this assessment to review what you’ve learned. It will not count toward your grade.
5 of 55 of 5 Items
Question
Use the image to answer the question.
A waveform is graphed on a coordinate plane.
Write the equation of the function graphed using the cosine.
(2 points)
f(x)=
cos
(
)+
Complete this assessment to review what you’ve learned. It will not count toward your grade.
5 of 55 of 5 Items
Question
Use the image to answer the question.
A waveform is graphed on a coordinate plane.
Write the equation of the function graphed using the cosine.
(2 points)
f(x)=
cos
(
)+
Answers
GPT-4o mini
To write the equation of the periodic function using the cosine function, we need to determine the amplitude, period, phase shift, and vertical shift (if any).
1. **Amplitude**: This is the height of the wave from its midline to a peak. Measure the maximum and minimum values of the function and use the formula:
\[
\text{Amplitude} = \frac{\text{Max value} - \text{Min value}}{2}
\]
2. **Period**: This is the distance required for the waveform to complete one full cycle. Measure the x-distance between two consecutive peaks (or troughs).
3. **Phase Shift**: This is the horizontal shift from the origin. If the waveform begins at a point other than the peak at \( x = 0 \), calculate this shift.
4. **Vertical Shift**: If the midline of the waveform is above or below the x-axis, measure this value.
Using all of this information, the general equation of cosine functions can be given by:
\[
f(x) = A \cos(B(x - C)) + D
\]
Where:
- \( A \) is the amplitude,
- \( B \) is related to the period (\( B = \frac{2\pi}{\text{Period}} \)),
- \( C \) is the phase shift,
- \( D \) is the vertical shift.
Once you provide the graph details (like amplitude, period, phase shift, etc.) or their numerical values, I can help you write the specific equation for the function.
1. **Amplitude**: This is the height of the wave from its midline to a peak. Measure the maximum and minimum values of the function and use the formula:
\[
\text{Amplitude} = \frac{\text{Max value} - \text{Min value}}{2}
\]
2. **Period**: This is the distance required for the waveform to complete one full cycle. Measure the x-distance between two consecutive peaks (or troughs).
3. **Phase Shift**: This is the horizontal shift from the origin. If the waveform begins at a point other than the peak at \( x = 0 \), calculate this shift.
4. **Vertical Shift**: If the midline of the waveform is above or below the x-axis, measure this value.
Using all of this information, the general equation of cosine functions can be given by:
\[
f(x) = A \cos(B(x - C)) + D
\]
Where:
- \( A \) is the amplitude,
- \( B \) is related to the period (\( B = \frac{2\pi}{\text{Period}} \)),
- \( C \) is the phase shift,
- \( D \) is the vertical shift.
Once you provide the graph details (like amplitude, period, phase shift, etc.) or their numerical values, I can help you write the specific equation for the function.