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
