To find the period and amplitude of the function [tex]\( y = 4 \sin 6x \)[/tex], let's follow these steps:
1. Identify the amplitude:
- The amplitude of a sine function [tex]\( y = A \sin(Bx) \)[/tex] is given by the absolute value of the coefficient of the sine function.
- In the given function [tex]\( y = 4 \sin 6x \)[/tex], the coefficient of the sine term is 4.
- Therefore, the amplitude is [tex]\( \boxed{4} \)[/tex].
2. Determine the period:
- The period of a sine function [tex]\( y = A \sin(Bx) \)[/tex] is calculated using the formula:
[tex]\[
\text{Period} = \frac{2\pi}{B}
\][/tex]
- In the given function [tex]\( y = 4 \sin 6x \)[/tex], the coefficient [tex]\( B \)[/tex] in front of [tex]\( x \)[/tex] is 6.
- Substituting [tex]\( B = 6 \)[/tex] into the period formula, we get:
[tex]\[
\text{Period} = \frac{2\pi}{6} = \frac{\pi}{3}
\][/tex]
- Thus, the exact value of the period is [tex]\( \boxed{\frac{\pi}{3}} \)[/tex].
So, the exact values are:
- Amplitude: [tex]\( \boxed{4} \)[/tex]
- Period: [tex]\( \boxed{\frac{\pi}{3}} \)[/tex]
