Problem 24 - Algebra and Trigonometry

Question

Find the amplitude and period of the function, and sketch its graph. $$ y=-3 \sin \pi x $$

Step-by-Step Solution

Answer

Amplitude is 3, period is 2; the graph is an inverted sine wave.

Step 1: Identify the General Form of the Sinusoidal Function

The input function is given as \( y = -3 \sin(\pi x) \). The general form for a sinusoidal function is \( y = A \sin(Bx + C) + D \). Here, we identify \( A = -3 \), \( B = \pi \), \( C = 0 \), and \( D = 0 \).

Step 2: Find the Amplitude

The amplitude of a sinusoidal function \( y = A \sin(Bx + C) + D \) is given by the absolute value of \( A \). For this function, \( A = -3 \), so the amplitude is \( |A| = 3 \).

Step 3: Find the Period

The period of a sinusoidal function \( y = A \sin(Bx + C) + D \) is calculated using \( \frac{2\pi}{|B|} \). Here, \( B = \pi \), so the period is \( \frac{2\pi}{\pi} = 2 \).

Step 4: Sketch the Graph

Use the amplitude and period to sketch the graph. The graph of \( y = -3 \sin(\pi x) \) is a sine wave (inverted due to the negative amplitude) with an amplitude of 3 and a period of 2. It oscillates between -3 and 3. One full cycle of the sine wave is completed as \( x \) goes from 0 to 2.

Key Concepts

Understanding Amplitude

Amplitude is a key characteristic of sinusoidal functions like sine and cosine waves. It represents the height of the wave from its central axis to the peak or trough. Think of amplitude as the measure of the wave's strength or intensity.

In the case of the function given in the exercise, \( y = -3 \sin(\pi x) \), the amplitude is determined by the coefficient of the sine function, which is \( A \). Here, \( A = -3 \). Since amplitude is always a positive quantity, we take the absolute value: \(|A| = 3\).

So, the amplitude tells us how far the wave stretches vertically, here it is 3 units away from the horizontal axis either upwards or downwards. In graphical terms, this means the wave will oscillate between -3 and 3. In this function, the negative sign in \( -3 \) indicates the wave is flipped vertically, but it doesn't affect the amplitude itself.

The Concept of Period

The period of a trigonometric function is the distance it takes for the function to complete one full cycle and start repeating its pattern. For sinusoidal functions, the formula to find the period is \( \frac{2\pi}{|B|} \), where \( B \) is the coefficient inside the sine function that affects the horizontal stretch or compression.

In the given function \( y = -3 \sin(\pi x) \), \( B \) is \( \pi \). Plugging this into the formula, we get the period as \( \frac{2\pi}{\pi} = 2 \).

This value of 2 means the wave completes a full cycle every 2 units along the x-axis. If you were to graph this function, you would see the wave start at one point, travel up, then down, and back to start in exactly that span. Period is crucial for understanding how quickly the wave repeats, and it directly impacts the shape of the graph.

Introduction to Sinusoidal Functions

Sinusoidal functions encompass both sine and cosine waves, which are fundamental in trigonometry. These functions are periodic, meaning they repeat in regular intervals.

The general form of a sinusoidal function is \( y = A \sin(Bx + C) + D \). Each parameter has a distinct role:

• \( A \) determines the amplitude.
• \( B \) is related to the period of the wave.
• \( C \) causes horizontal shifts, moving the wave left or right.
• \( D \) causes vertical shifts, moving the wave up or down.

In our given function \( y = -3 \sin(\pi x) \), we have identified \( A = -3 \), \( B = \pi \), and both \( C \) and \( D \) as 0. This tells us the wave has an amplitude of 3, a period of 2, and no horizontal or vertical shifts.

Overall, understanding how each component affects the function is vital for analyzing and graphing sinusoidal waves effectively.