Question

Which functions have an amplitude of 4 and a period of 2 ? (A) \(y=4 \cos 2 x\) (B) \(y=-4 \sin \pi x\) (C) \(y=2 \sin 4 x\) (D) \(y=4 \cos \pi x\)

Short answer

The functions that have an amplitude of 4 and a period of 2 are \(y=-4 \sin \pi x\) and \(y=4 \cos \pi x\).

Step-by-step solution

Identify the amplitude

Check the absolute value of the coefficient for each function: for \((A)\) \(4\), for \((B)\) \(4\), for \((C)\) \(2\) and for \((D)\) \(4\). Therefore, functions \((A)\), \((B)\), and \((D)\) have the amplitude of \(4\). Now let's check which of these have a period equal to \(2\).

Identify the period

To do this, use the formula for the period of trigonometric functions: \( \frac{2\pi}{\vert B \vert} \). We need this to be equal to \(2\). Hence, \( \frac{2\pi}{\vert B \vert} = 2 \Rightarrow B = \pi \). Check the coefficient of \(x\) in each function for this value of \(B\): for \((A)\) \(2\), for \((B)\) \(\pi\) and for \((D)\) \(\pi\). Therefore, only functions \(B\) and \(D\) have a period of \(2\).

Final selection

Now we know that the functions that satisfy both conditions (amplitude of \(4\) and period of \(2\)) are the ones that have a \(4\) as coefficient and a \(\pi\) as coefficient of \(x\), which are option \((B)\) \(y=-4 \sin \pi x\) and option \((D)\) \(y=4 \cos \pi x\).

Key concepts

Amplitude

Amplitude is an essential concept when studying trigonometric functions. It tells us how "tall" or "short" a wave is by measuring the distance from the wave's midpoint to its peak or trough.
In other words, amplitude indicates the maximum disturbance from the average value in a periodic function such as a sine or cosine wave.

If you visualize a trigonometric wave, think of amplitude as how far the curve goes up or down from the middle.
To get the amplitude, you look at the coefficient in front of the trigonometric function.

• If we see something like \(4 \cos x\), then the amplitude is \(4\).
• The amplitude is always a positive number, so if the coefficient were negative, like \(-4 \cos x\), it would still just be \(4\).

Understanding the amplitude helps us comprehend how intense or weak the oscillations of the function are.

Period

The period of a trigonometric function is another vital characteristic. It represents how "wide" the wave is, or how far along the horizontal axis the wave must go before repeating its pattern.
The period tells us the length of one complete cycle:

For sine and cosine functions, the formula for period is \(\frac{2\pi}{\vert B \vert}\), where \(B\) is the coefficient of \(x\) in the function.

• For example, if we have \(y = \cos 2x\), the period would be \(\frac{2\pi}{2} = \pi\).
• If the equation were \(y = \cos \pi x\), the period would be \(\frac{2\pi}{\pi} = 2\).

This understanding helps differentiate between quickly and slowly repeating functions, which is crucial in determining the behavior of trigonometric functions.

Cosine Function

The cosine function, along with sine, is one of the most fundamental trigonometric functions. It forms the basis of many waves and patterns that we see in math and physics.
The general form of a cosine function is given by \(y = A \cos(Bx + C) + D\).

When you graph it, the cosine wave starts from its maximum point because cosine of zero is 1.
Some essential elements to keep in mind about cosine functions include:

• The amplitude (\(A\)) dictates the wave's height.
• The period (\(\frac{2\pi}{\vert B \vert}\)) influences how wide each cycle is.
• Phase shift (\