Question
Periodic Functions Online Practice Part 2
Complete this assessment to review what you’ve learned. It will not count toward your grade.
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Question
Identify the transformations for the function f(x)=−3cos(4x−π2)+2
.(1 point)
Responses
vertical stretch by a factor of −3
, horizontal compression by 14
, horizontal shift right by π2
, and vertical shift up 2
vertical stretch by a factor of negative 3 , horizontal compression by Start Fraction 1 over 4 End Fraction , horizontal shift right by Start Fraction pi over 2 End Fraction , and vertical shift up 2
reflection across the x-axis, vertical stretch by a factor of 3, horizontal stretch by 4, horizontal shift right by π8
, and vertical shift up 2
reflection across the x -axis, vertical stretch by a factor of 3, horizontal stretch by 4, horizontal shift right by Start Fraction pi over 8 End Fraction , and vertical shift up 2
reflection across the x-axis, vertical stretch by a factor of 3, horizontal compression by 14
, horizontal shift right by π8
, and vertical shift up 2
reflection across the x -axis, vertical stretch by a factor of 3, horizontal compression by Start Fraction 1 over 4 End Fraction , horizontal shift right by Start Fraction pi over 8 End Fraction , and vertical shift up 2
vertical stretch by a factor of −3
, horizontal compression by 14
, horizontal shift right by π8
, and vertical shift up 2
vertical stretch by a factor of negative 3 , horizontal compression by Start Fraction 1 over 4 End Fraction , horizontal shift right by Start Fraction pi over 8 End Fraction , and vertical shift up 2
Remaining Attempts : 3
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Answers
GPT-4o mini
To analyze the function \( f(x) = -3\cos(4x - \frac{\pi}{2}) + 2 \) and identify the transformations, we should break down each part of the equation:
1. **Reflection Across the X-Axis**: The negative sign in front of the 3 indicates a reflection across the x-axis.
2. **Vertical Stretch**: The factor of 3 (ignoring the negative sign) indicates a vertical stretch by a factor of 3.
3. **Horizontal Compression**: The coefficient of 4 inside the cosine function indicates a horizontal compression by a factor of \( \frac{1}{4} \) (since you divide the angle by this factor).
4. **Horizontal Shift**: The term \( 4x - \frac{\pi}{2} \) can be rewritten as \( 4(x - \frac{\pi}{8}) \) when factoring out the 4. This indicates a horizontal shift to the right by \( \frac{\pi}{8} \).
5. **Vertical Shift**: The +2 indicates a vertical shift upwards by 2.
Now, putting this all together, the transformations are:
- Reflection across the x-axis
- Vertical stretch by a factor of 3
- Horizontal compression by a factor of \( \frac{1}{4} \)
- Horizontal shift right by \( \frac{\pi}{8} \)
- Vertical shift up by 2
With this breakdown, the most fitting response among the provided options is:
**Reflection across the x-axis, vertical stretch by a factor of 3, horizontal compression by \( \frac{1}{4} \), horizontal shift right by \( \frac{\pi}{8} \), and vertical shift up by 2.**
1. **Reflection Across the X-Axis**: The negative sign in front of the 3 indicates a reflection across the x-axis.
2. **Vertical Stretch**: The factor of 3 (ignoring the negative sign) indicates a vertical stretch by a factor of 3.
3. **Horizontal Compression**: The coefficient of 4 inside the cosine function indicates a horizontal compression by a factor of \( \frac{1}{4} \) (since you divide the angle by this factor).
4. **Horizontal Shift**: The term \( 4x - \frac{\pi}{2} \) can be rewritten as \( 4(x - \frac{\pi}{8}) \) when factoring out the 4. This indicates a horizontal shift to the right by \( \frac{\pi}{8} \).
5. **Vertical Shift**: The +2 indicates a vertical shift upwards by 2.
Now, putting this all together, the transformations are:
- Reflection across the x-axis
- Vertical stretch by a factor of 3
- Horizontal compression by a factor of \( \frac{1}{4} \)
- Horizontal shift right by \( \frac{\pi}{8} \)
- Vertical shift up by 2
With this breakdown, the most fitting response among the provided options is:
**Reflection across the x-axis, vertical stretch by a factor of 3, horizontal compression by \( \frac{1}{4} \), horizontal shift right by \( \frac{\pi}{8} \), and vertical shift up by 2.**