To determine the characteristics of the function [tex]\( f(x) = 3 \sin \left(x - \frac{\pi}{2}\right) + 1 \)[/tex], we need to analyze the given statements using the information provided in the table and characteristics of the sine function:
1. The function has a midline at [tex]\( y = 0 \)[/tex]:
To find the midline of the function [tex]\( f(x) \)[/tex], we observe that it is of the form [tex]\( f(x) = A \sin(Bx - C) + D \)[/tex] where [tex]\( D \)[/tex] is the vertical shift or midline. Here, [tex]\( D = 1 \)[/tex]. Thus, the midline of the function is at [tex]\( y = 1 \)[/tex], not at [tex]\( y = 0 \)[/tex]. This statement is false.
2. The function has a period of [tex]\( 2\pi \)[/tex]:
The sine function has a period of [tex]\( 2\pi \)[/tex]. Since the function [tex]\( f(x) \)[/tex] is a sine function that has been horizontally shifted and vertically translated, but not scaled horizontally, the period remains [tex]\( 2\pi \)[/tex]. This statement is true.
3. The function has a minimum value of -2:
From the table, we can see that the minimum value of [tex]\( f(x) \)[/tex] is -2 when [tex]\( x = 0 \)[/tex]. This statement is true.
4. The function has a maximum value of 4:
According to the table, the maximum value of [tex]\( f(x) \)[/tex] is 4 (at [tex]\( x = -\pi \)[/tex] and [tex]\( x = \pi \)[/tex]). This statement is true.
5. The function is decreasing on the interval [tex]\( \left( 0, \frac{\pi}{2} \right) \)[/tex]:
To check if the function is decreasing on the interval [tex]\( \left(0, \frac{\pi}{2}\right) \)[/tex]:
- [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex] is [tex]\( -2 \)[/tex].
- [tex]\( f(x) \)[/tex] at [tex]\( x = \frac{\pi}{2} \)[/tex] is [tex]\( 1 \)[/tex].
Since [tex]\( f(x) \)[/tex] increases from -2 to 1 over the interval [tex]\( \left(0, \frac{\pi}{2}\right) \)[/tex], it is increasing rather than decreasing on this interval. This statement is false.
Thus, the true statements about the function [tex]\( f \)[/tex] are:
- The function has a period of [tex]\( 2\pi \)[/tex].
- The function has a minimum value of -2.
- The function has a maximum value of 4.
