Question
Periodic Functions Online Practice Part 2
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Question
Use the image to answer the question.
A sinusoidal wave is drawn in a coordinate plane. The x-axis ranges from negative pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 3 to 3 in increments of 1.
Determine the equation of the sine function shown in the graph.
(1 point)
Responses
f(x)=sinx+π2
f left parenthesis x right parenthesis equals sine x plus Start Fraction pi over 2 End Fraction
f(x)=sin(x−π2)
f left parenthesis x right parenthesis equals sine left parenthesis x minus Start Fraction pi over 2 End Fraction right parenthesis
f(x)=sin(x+π2)
f left parenthesis x right parenthesis equals sine left parenthesis x plus Start Fraction pi over 2 End Fraction right parenthesis
f(x)=sinx−π2
f left parenthesis x right parenthesis equals sine x minus Start Fraction pi over 2 End Fraction
Remaining Attempts : 3
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Answers
GPT-4o mini
To determine the equation of the sine function from the graph you described, we need to consider a few key features of sine functions:
1. **Amplitude**: The amplitude is the distance from the midline to the peak (or trough) of the wave. Given that the y-axis ranges from -3 to 3, the amplitude is 3.
2. **Vertical Shift**: The midline of the function can be found midway between the maximum and minimum values. In this case, the midline is at y = 0, which means there is no vertical shift.
3. **Period**: The period of a sine function is the distance along the x-axis for one complete cycle of the wave. For a typical sine function, the period is \(2\pi\). If the wave in the graph does go through one full cycle from \(0\) to \(2\pi\), the period is maintained.
4. **Phase Shift**: To determine any phase shift, identify where the function starts in relation to a standard sine function. The standard sine function \(f(x) = \sin(x)\) starts at (0, 0) and moves to the maximum at \(\frac{\pi}{2}\). Any horizontal shift will affect the function.
Let's analyze the provided choices based on these factors:
- **f(x) = sin(x + π/2)**: This would represent a sine wave that starts at the peak (maximum) because adding \(\frac{\pi}{2}\) to x shifts the standard sine function left by \(\frac{\pi}{2}\), starting it at a high point.
- **f(x) = sin(x − π/2)**: This would shift the standard sine function to the right. It would start at a low point instead of the midline.
- **f(x) = sin(x) + π/2**: This would vertically shift the sine function upward, which isn't consistent here.
- **f(x) = sin(x) − π/2**: This would also shift the graph downward.
Given that you have indicated vertical ranges and the nature of sine waves likely starting in a middle position at \(y=0\), the most fitting option is **f(x) = sin(x + π/2)**, indicating it starts at a peak from what you're describing.
Thus, the answer is:
**f(x) = sin(x + π/2)**
1. **Amplitude**: The amplitude is the distance from the midline to the peak (or trough) of the wave. Given that the y-axis ranges from -3 to 3, the amplitude is 3.
2. **Vertical Shift**: The midline of the function can be found midway between the maximum and minimum values. In this case, the midline is at y = 0, which means there is no vertical shift.
3. **Period**: The period of a sine function is the distance along the x-axis for one complete cycle of the wave. For a typical sine function, the period is \(2\pi\). If the wave in the graph does go through one full cycle from \(0\) to \(2\pi\), the period is maintained.
4. **Phase Shift**: To determine any phase shift, identify where the function starts in relation to a standard sine function. The standard sine function \(f(x) = \sin(x)\) starts at (0, 0) and moves to the maximum at \(\frac{\pi}{2}\). Any horizontal shift will affect the function.
Let's analyze the provided choices based on these factors:
- **f(x) = sin(x + π/2)**: This would represent a sine wave that starts at the peak (maximum) because adding \(\frac{\pi}{2}\) to x shifts the standard sine function left by \(\frac{\pi}{2}\), starting it at a high point.
- **f(x) = sin(x − π/2)**: This would shift the standard sine function to the right. It would start at a low point instead of the midline.
- **f(x) = sin(x) + π/2**: This would vertically shift the sine function upward, which isn't consistent here.
- **f(x) = sin(x) − π/2**: This would also shift the graph downward.
Given that you have indicated vertical ranges and the nature of sine waves likely starting in a middle position at \(y=0\), the most fitting option is **f(x) = sin(x + π/2)**, indicating it starts at a peak from what you're describing.
Thus, the answer is:
**f(x) = sin(x + π/2)**