Question

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

Explanation:

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

The general form of a sine function is \(y = A\sin(Bx - C)+D\). The period \(T\) is given by the formula \(T=\frac{2\pi}{|B|}\).
For the function \(y=-4\sin(5\pi x)\), we have \(B = 5\pi\).

Step2: Calculate the period

Substitute \(B = 5\pi\) into the period formula \(T=\frac{2\pi}{|B|}\).
Since \(B = 5\pi\), then \(T=\frac{2\pi}{5\pi}\).
Cancel out the \(\pi\) terms: \(T=\frac{2}{5}\).

Step3: Recall the formula for the amplitude of a sine function

The general form of a sine function is \(y = A\sin(Bx - C)+D\). The amplitude \(|A|\) (the distance from the mid - line of the sine wave to its maximum or minimum value).
For the function \(y=-4\sin(5\pi x)\), we have \(A=-4\).

Step4: Calculate the amplitude

The amplitude is \(|A|\). Since \(A = - 4\), then \(|A|=|-4| = 4\).

Step5: Analyze the graph

The standard sine function \(y=\sin(x)\) has a mid - line \(y = 0\). The function \(y=-4\sin(5\pi x)\) has an amplitude of \(4\), so the maximum value is \(y = 4\) and the minimum value is \(y=-4\). The negative sign in front of the \(4\) reflects the graph of \(y = \sin(x)\) over the \(x\) - axis.

Answer:

• Period: \(\frac{2}{5}\)
• Amplitude: \(4\)
• Graph: Option A (assuming the graph in option A has an amplitude of \(4\) (ranges from \(y=-4\) to \(y = 4\)) and a period of \(\frac{2}{5}\))