Determine the period of the function [tex]y = -3 \cos \left(\frac{\pi}{5} x\right)[/tex].

A. 3
B. 8
C. -3
D. 10

Please select the best answer from the choices provided:
A, B, C, or D



Answer :

To determine the period of the function [tex]\( y = -3 \cos \left( \frac{\pi}{5} x \right) \)[/tex], we need to understand the general form of a cosine function and how its period is calculated.

The general form of a cosine function is:
[tex]\[ y = a \cos(bx + c) + d \][/tex]

The period of this function is given by:
[tex]\[ \text{Period} = \frac{2\pi}{|b|} \][/tex]

In the given function [tex]\( y = -3 \cos \left( \frac{\pi}{5} x \right) \)[/tex], we can identify the coefficient [tex]\( b \)[/tex]:
[tex]\[ b = \frac{\pi}{5} \][/tex]

Now, we use the formula for the period:
[tex]\[ \text{Period} = \frac{2\pi}{\left| \frac{\pi}{5} \right|} \][/tex]

Simplifying this expression:
[tex]\[ \text{Period} = \frac{2\pi}{\frac{\pi}{5}} = 2\pi \times \frac{5}{\pi} = 2 \times 5 = 10 \][/tex]

So, the period of the function [tex]\( y = -3 \cos \left( \frac{\pi}{5} x \right) \)[/tex] is 10.

Therefore, the correct answer is:
- d. 10