Question

Determine the amplitude and period of each function without graphing. $$ y=\frac{9}{5} \cos \left(-\frac{3 \pi}{2} x\right) $$

Short answer

Amplitude: \frac{9}{5}\, Period: \frac{4}{3}\.

Step-by-step solution

- Identify the Amplitude

The amplitude of a cosine function of the form \(y = a \cos(bx)\) is given by the absolute value of \a\. For the function \y = \frac{9}{5} \cos \left(-\frac{3 \pi}{2} x\right)\, the amplitude is \left| \frac{9}{5} \right| = \frac{9}{5}\.

- Identify the Period

The period of a cosine function \(y = \cos(bx)\) is given by \[ \text{Period} = \frac{2 \pi}{|b|} \]. Here, \(b = -\frac{3 \pi}{2}\). We only consider the absolute value of \b\, so \left| -\frac{3 \pi}{2} \right| = \frac{3 \pi}{2}\. Therefore, the period of the function is: \text{Period} = \frac{2 \pi}{\frac{3 \pi}{2}} = \frac{2 \pi \times 2}{3 \pi} = \frac{4}{3}\.

Key concepts

Amplitude

The amplitude of a trigonometric function, particularly a cosine function, measures how much the function oscillates vertically from its central position. For a general cosine function of the form \(y = a \text{cos}(bx)\) , the amplitude is given by the absolute value of the coefficient \(a\) . The amplitude will always be positive and tells us the peak distance from the centerline of the graph.
In the exercise, with the function \(y = \frac{9}{5} \text{cos} \big( -\frac{3 \text{π}}{2} x \big)\) , the coefficient \(a\) is \(\frac{9}{5}\) .
Thus, the amplitude is \(\big| \frac{9}{5} \big| = \frac{9}{5}\) . This means the graph of our function will reach up to \(\frac{9}{5}\) units above and \(\frac{9}{5}\) units below the central axis.

Period

The period of a trigonometric function describes the length of one complete cycle of the wave before it starts to repeat. For a cosine function of the form \(y = \text{cos}(bx)\) , the period is determined by the coefficient \(b\) and is calculated using the formula: \( \text{Period} = \frac{2 \text{π}}{|b|} \) .
In our case, the function is \(y = \frac{9}{5} \text{cos} \big( -\frac{3 \text{π}}{2} x \big)\) and \(b = -\frac{3 \text{π}}{2}\) . Notice that we're taking the absolute value of \(b\) , which nullifies the negative sign.
Therefore, the period is computed as: \(\text{Period} = \frac{2 \text{π}}{\frac{3 \text{π}}{2}} = \frac{2 \text{π} \times 2}{3 \text{π}} = \frac{4}{3}\). This means that every \(\frac{4}{3}\) units along the x-axis, the pattern of the cosine function repeats.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is periodic, meaning it repeats its values in regular intervals or periods. The general form of the cosine function is \(y = a \text{cos}(bx)\) , where:

• \(a\) controls the amplitude.
• \(b\) affects the period.

The cosine function oscillates between \(-a\) and \(a\) . The function’s maximum value is \(a\) and its minimum value is \(-a\) . The standard period of \(\text{cos}(x)\) is \(2 \text{π}\) , but when we introduce the coefficient \(b\), the period changes to \(\frac{2 \text{π}}{|b|}\) .
In the given problem, we examined \(y = \frac{9}{5} \text{cos} \big( -\frac{3 \text{π}}{2} x \big)\) . The coefficient \(a\) determines the amplitude as \(\frac{9}{5}\) , while the coefficient \(b\) specifies the period length as \(\frac{4}{3}\) units along the x-axis. Understanding these properties allows you to sketch the wave's shape and predict its behavior without plotting it point-by-point.