Question

What is the period of LaTeX: y = -3\sin\left(\pi{x}+\frac{\pi}{2}\right)-9.

Answer 1

To find the period of the function \( y = -3\sin\left(\pi{x}+\frac{\pi}{2}\right)-9 \), we start by identifying the essential part of the sine function.

The general form of a sine function is given by:

\[
y = a \sin(bx + c) + d
\]

where:
- \( a \) is the amplitude,
- \( b \) affects the period,
- \( c \) is the phase shift,
- \( d \) is the vertical shift.

For the function \( y = -3\sin\left(\pi{x}+\frac{\pi}{2}\right)-9 \):
- The coefficient \( b \) is \( \pi \).

The period \( P \) of the sine function can be computed using the formula:

\[
P = \frac{2\pi}{|b|}
\]

Substituting \( b = \pi \) into the formula:

\[
P = \frac{2\pi}{|\pi|} = \frac{2\pi}{\pi} = 2
\]

Thus, the period of the function \( y = -3\sin\left(\pi{x}+\frac{\pi}{2}\right)-9 \) is

\[
\boxed{2}
\]