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The table shows points on the graph of the function [tex]f(x) = 3 \sin \left( x - \frac{\pi}{2} \right) + 1[/tex].

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$-\frac{3\pi}{2}$[/tex] & [tex]$-\pi$[/tex] & [tex]$-\frac{\pi}{2}$[/tex] & 0 & [tex]$\frac{\pi}{2}$[/tex] & [tex]$\pi$[/tex] & [tex]$\frac{3\pi}{2}$[/tex] \\
\hline
[tex]$f(x)$[/tex] & 1 & 4 & 1 & -2 & 1 & 4 & 1 \\
\hline
\end{tabular}

Which statements are true about function [tex]f[/tex]?

- The function has a midline at [tex]y = 0[/tex].
- The function has a period of [tex]2\pi[/tex].
- The function has a minimum value of [tex]-2[/tex].
- The function has a maximum value of [tex]4[/tex].
- The function is decreasing on the interval [tex]\left(0, \frac{\pi}{2}\right)[/tex].

Sagot :

GinnyAnsweridn
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.
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