Question

Determine the amplitude and period of each function without graphing. $$ y=-3 \cos (4 x) $$

Short answer

Amplitude: 3. Period: \ \frac{\raisebox{0.7ex}{1}}{2} \pi \.

Step-by-step solution

Identify the amplitude

The amplitude of a cosine function in the form of \[ y=a \, \cos(bx) \] is given by the absolute value of the coefficient \a\. For the function \[ y=-3 \cos (4 x) \], \a=-3\. Thus, the amplitude is \|a| = |-3| = 3\.

Determine the period

The period of a cosine function in the form of \[ y=a \cos(bx) \] is given by \frac{2\pi }{b}\. For the function \ y=-3 \cos (4 x)\, \ b=4\. Hence, the period is \ \frac{2 \pi}{4} = \ \frac{\raisebox{0.7ex}{1}}{2} \pi \.

Key concepts

Amplitude

The amplitude of a trigonometric function describes how far the function's values reach from its central position, up or down. Imagine the midline as the average value of the function. The amplitude measures how 'tall' or 'deep' the waves of the function are from that midline.
For the function \(y = -3 \cos (4x)\), the amplitude is found by looking at the coefficient in front of the cosine term. In general, for \(y = a \cos(bx)\), the amplitude is \|a|\, which means the absolute value of \a\.
So, in the case of \y = -3 \cos (4x)\, the coefficient \a\ is -3. The absolute value of -3 is 3, so the amplitude is 3. This means the function's values swing from 3 units above the midline to 3 units below it.

Period

The period of a trigonometric function is the distance (along the x-axis) it takes for the function to complete one full cycle. For example, if you start at one peak of a cosine wave, the period is the horizontal distance to the next peak.
For the cosine function \(y = a \cos(bx)\), the period is calculated by dividing \2\pi\ by the coefficient \b\. This coefficient affects how many cycles fit into a given length along the x-axis.
If we look at \y = -3 \cos (4x)\, the coefficient \b\ is 4. Thus, the period \[ P \] can be found using the formula:
\frac{2\pi}{b}\Substituting \b = 4\, we get:
\frac{2\pi}{4} = \frac{\raisebox{0.7ex}{1}}{2}\pi\So, the period is \frac{\pi}{2}\, meaning the function repeats every \frac{\raisebox{0.7ex}{1}}{2}\pi \ units along the x-axis.

Cosine function

Trigonometric functions like sine and cosine are fundamental in understanding periodic phenomena, which are patterns that repeat over intervals. The cosine function, often written as \(y = \cos(x)\), is especially useful.
The cosine function generates a wave that starts at a peak when \x = 0\. It oscillates between 1 and -1, forming peaks and troughs at regular intervals. This oscillation is why we call it a periodic function.
By adjusting its amplitude and period via coefficients \a \ and \b\, we can shape the cosine wave for various applications:

• The amplitude \|a|\ determines how high or low the wave goes from its midline.
• The period \frac{2\pi}{b}\ tells us how long it takes to complete one cycle.

Such trigonometric adjustments find use in fields like physics, engineering, and signal processing.
In \y = -3 \cos (4x)\, the coefficient -3 changes the amplitude to 3, while the coefficient 4 compresses the period to \frac{\pi}{2}\. This is a testament to the practical flexibility of trigonometric functions.