Question
Periodic Function Flexibility Quick Check
4 of 54 of 5 Items
Question
Use the image to answer the question.
A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from 0 to 2 in increments of 0.5.
Which of the following options correctly writes an equation for a cosine function, based on the given graph?
(1 point)
Responses
y=cos(x)+1
y equals cosine left parenthesis x right parenthesis plus 1
y=−cos(x)
y eqauls negative cosine left parenthesis x right parenthesis
y=−cos(x)+1
y equals negative cosine left parenthesis x right parenthesis plus 1
y=−cos(x−1)+1
y equals negative cosine left parenthesis x minus 1 right parenthesis plus 1
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Answers
GPT-4o mini
To determine the correct equation for the cosine function based on the provided description of the graph, we need to analyze the key features:
1. **Amplitude**: The cosine function's height typically ranges from -1 to 1, but the question indicates that the y-axis runs from 0 to 2. This suggests a vertical transformation has occurred, likely indicating a vertical shift.
2. **Vertical Shift**: If the cosine function is shifted upwards, it indicates a vertical translation. A cosine function that oscillates between 0 and 2 could be indicated by a vertical shift of +1.
3. **Reflection**: If the cosine wave is inverted (flipped across the x-axis), this would be indicated by a negative sign in front of the cosine function.
4. **Phase Shift**: If the graph is shifted horizontally, this can be represented as \(x - c\) or \(x + c\), where \(c\) is the phase shift.
Given these transformations:
- The options provided have different combinations of vertical and horizontal shifts and reflections.
Now, evaluating each response based on a typical cosine function \(y = \cos(x)\):
1. **y = cos(x) + 1**: This indicates a vertical shift up by 1. The range would be from 0 to 2.
2. **y = -cos(x)**: This represents a reflection across the x-axis with no vertical shift. The range would be from -1 to 1.
3. **y = -cos(x) + 1**: This indicates a reflection across the x-axis and a vertical shift up by 1. The range would be from 0 to 2.
4. **y = -cos(x - 1) + 1**: This would represent a cosine function reflected across the x-axis, shifted right by 1 unit, and vertically shifted up by 1. This still oscillates between 0 to 2.
Based on the possible interpretations:
- **If the graph is oscillating between 0 and 2**, then the most plausible equations are:
- \(y = \cos(x) + 1\)
- \(y = -\cos(x) + 1\)
If the cosine function in the image appears inverted and shifted upwards, then **the correct equation would be**:
**y = -cos(x) + 1**
This option correctly shows the wave may reflect down and still oscillate between 0 and 2. You may confirm the visual appearance to ensure it matches with one of the characteristics described.
1. **Amplitude**: The cosine function's height typically ranges from -1 to 1, but the question indicates that the y-axis runs from 0 to 2. This suggests a vertical transformation has occurred, likely indicating a vertical shift.
2. **Vertical Shift**: If the cosine function is shifted upwards, it indicates a vertical translation. A cosine function that oscillates between 0 and 2 could be indicated by a vertical shift of +1.
3. **Reflection**: If the cosine wave is inverted (flipped across the x-axis), this would be indicated by a negative sign in front of the cosine function.
4. **Phase Shift**: If the graph is shifted horizontally, this can be represented as \(x - c\) or \(x + c\), where \(c\) is the phase shift.
Given these transformations:
- The options provided have different combinations of vertical and horizontal shifts and reflections.
Now, evaluating each response based on a typical cosine function \(y = \cos(x)\):
1. **y = cos(x) + 1**: This indicates a vertical shift up by 1. The range would be from 0 to 2.
2. **y = -cos(x)**: This represents a reflection across the x-axis with no vertical shift. The range would be from -1 to 1.
3. **y = -cos(x) + 1**: This indicates a reflection across the x-axis and a vertical shift up by 1. The range would be from 0 to 2.
4. **y = -cos(x - 1) + 1**: This would represent a cosine function reflected across the x-axis, shifted right by 1 unit, and vertically shifted up by 1. This still oscillates between 0 to 2.
Based on the possible interpretations:
- **If the graph is oscillating between 0 and 2**, then the most plausible equations are:
- \(y = \cos(x) + 1\)
- \(y = -\cos(x) + 1\)
If the cosine function in the image appears inverted and shifted upwards, then **the correct equation would be**:
**y = -cos(x) + 1**
This option correctly shows the wave may reflect down and still oscillate between 0 and 2. You may confirm the visual appearance to ensure it matches with one of the characteristics described.