Question

y = - 4 sin 5πx
what is the period of the function y = - 4 sin 5πx?
(simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)
what is the amplitude of the function y = - 4 sin 5πx?
(type an integer or a simplified fraction.)
which of the following is the correct graph of the function y = - 4 sin 5πx?
a.
b.

Explanation:

Step1: Recall the formula for the period of a sine function

The general form of a sine function is \(y = A\sin(Bx + C)+D\), and its period \(T=\frac{2\pi}{|B|}\).

Step2: Identify \(B\) in the given function

For the function \(y=-4\sin(5\pi x)\), \(B = 5\pi\).

Step3: Calculate the period

Substitute \(B = 5\pi\) into the period formula \(T=\frac{2\pi}{|B|}\), we get \(T=\frac{2\pi}{5\pi}\).
Simplify \(\frac{2\pi}{5\pi}=\frac{2}{5}\).

Step4: Recall the formula for the amplitude of a sine function

The amplitude of the function \(y = A\sin(Bx + C)+D\) is \(|A|\).

Step5: Identify \(A\) in the given function

For the function \(y=-4\sin(5\pi x)\), \(A=-4\).

Step6: Calculate the amplitude

Substitute \(A = -4\) into the amplitude formula \(|A|\), we get \(| - 4|=4\).

Answer:

The period is \(\frac{2}{5}\), and the amplitude is \(4\).