Question
What are key feature of a sind parent function
Answers
GPT-4o mini
The key features of the sine parent function, typically denoted as \( f(x) = \sin(x) \), include the following:
1. **Domain**: The domain of the sine function is all real numbers, \( (-\infty, \infty) \).
2. **Range**: The range of the sine function is \([-1, 1]\). This means that the output values of the sine function will always fall within this interval.
3. **Period**: The sine function has a periodic behavior with a period of \( 2\pi \). This means that the function repeats its values every \( 2\pi \) radians.
4. **Amplitude**: The amplitude of the sine function is 1. The amplitude is the maximum distance from the midline (the horizontal line \( y = 0 \)) to the peak or trough of the wave.
5. **Key Points**: The sine function has specific key points within one period:
- \( f(0) = 0 \)
- \( f\left(\frac{\pi}{2}\right) = 1 \)
- \( f(\pi) = 0 \)
- \( f\left(\frac{3\pi}{2}\right) = -1 \)
- \( f(2\pi) = 0 \)
6. **Symmetry**: The sine function is an odd function, meaning it has symmetry about the origin. Mathematically, this is expressed as \( f(-x) = -f(x) \).
7. **Intersections with the Axes**: The sine function intersects the x-axis at integer multiples of \( \pi \), i.e., \( x = n\pi \) where \( n \) is an integer. It intersects the y-axis at the origin, (0, 0).
These features characterize the sine function and play a crucial role in the study of trigonometric functions, their transformations, and their applications in various fields such as physics, engineering, and signal processing.
1. **Domain**: The domain of the sine function is all real numbers, \( (-\infty, \infty) \).
2. **Range**: The range of the sine function is \([-1, 1]\). This means that the output values of the sine function will always fall within this interval.
3. **Period**: The sine function has a periodic behavior with a period of \( 2\pi \). This means that the function repeats its values every \( 2\pi \) radians.
4. **Amplitude**: The amplitude of the sine function is 1. The amplitude is the maximum distance from the midline (the horizontal line \( y = 0 \)) to the peak or trough of the wave.
5. **Key Points**: The sine function has specific key points within one period:
- \( f(0) = 0 \)
- \( f\left(\frac{\pi}{2}\right) = 1 \)
- \( f(\pi) = 0 \)
- \( f\left(\frac{3\pi}{2}\right) = -1 \)
- \( f(2\pi) = 0 \)
6. **Symmetry**: The sine function is an odd function, meaning it has symmetry about the origin. Mathematically, this is expressed as \( f(-x) = -f(x) \).
7. **Intersections with the Axes**: The sine function intersects the x-axis at integer multiples of \( \pi \), i.e., \( x = n\pi \) where \( n \) is an integer. It intersects the y-axis at the origin, (0, 0).
These features characterize the sine function and play a crucial role in the study of trigonometric functions, their transformations, and their applications in various fields such as physics, engineering, and signal processing.