Problem 33 - Algebra 2 and Trigonometry

Question

Sketch one cycle of each function. \(y=4 \sin 3 x\)

Step-by-Step Solution

Answer

Sketch one cycle starting at (0,0), peaking at \( \frac{\pi}{6} \), x-axis \( \frac{\pi}{3} \), trough at \( \frac{\pi}{2} \), ending at \( \frac{2\pi}{3} \).

Step 1: Understand the Function

The function given is a sine function in the form of \( y = A \sin(Bx) \). Here, \( A = 4 \), which represents the amplitude of the sine wave, and \( B = 3 \), which determines the frequency, affecting the period of the wave.

Step 2: Calculate the Period

The period of a sine function is calculated using the formula \( \frac{2\pi}{B} \). Substituting the given value of \( B = 3 \), we get \[ \text{Period} = \frac{2\pi}{3} \]. This tells us one full cycle of the sine wave completes over \( \frac{2\pi}{3} \) units along the x-axis.

Step 3: Identify Key Points

For one full cycle of the sine wave, identify the key points: - Start at \( (0, 0) \)- First peak at \( \left( \frac{\pi}{6}, 4 \right) \)- Cross the x-axis at \( \left( \frac{\pi}{3}, 0 \right) \)- First trough at \( \left( \frac{\pi}{2}, -4 \right) \)- End back at the x-axis at \( \left( \frac{2\pi}{3}, 0 \right) \)

Step 4: Plot the Cycle

Sketch the graph starting at the origin (0,0). Plot the key points identified: the first peak, the zero crossing, the trough, and the end of the cycle. Connect these points smoothly in a wave pattern typical for a sine wave, keeping in mind the amplitude of 4 and the nature of the sine function.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It arises from the coordinate system on a circle, where the sine of an angle is the y-coordinate of a point on the unit circle. A sine wave is smooth and periodic, reflecting the oscillatory nature of the function.

When expressed in the general form, it is written as \( y = A \sin(Bx) \), where:

• \( A \) is the amplitude, determining the vertical stretch or compression.
• \( B \) affects the period, controlling how quickly the wave completes a full cycle.

In examining the sine function, we see it oscillating between positive and negative values, crossing the origin at specific points, and it is renowned for its regular and predictable behavior.

Amplitude

Amplitude refers to the magnitude or height of the peaks and troughs of a sine wave. It is denoted by \( A \) in the sine function formula \( y = A \sin(Bx) \). The amplitude determines how "tall" or "short" the wave appears, extending equally above and below the central axis, typically the horizontal x-axis.

In the equation \(y = 4 \sin 3x\), the amplitude is 4, meaning the wave oscillates between the values +4 and -4.

• The amplitude gives us direct insight into how intense the oscillations of the wave are.
• It is always a positive value, since it represents distance and cannot be negative.

A larger amplitude results in a taller wave, while a smaller amplitude results in a flatter wave.

Period

The period of a sine function is the length it takes to complete one full cycle of its wave pattern. It is a crucial characteristic that defines the wave's frequency and repetition over a given domain, measured along the x-axis.

The formula for calculating the period of a sine function \(y = A \sin(Bx)\) is given by \( \frac{2\pi}{B} \).

For the function \(y = 4 \sin 3x\), the period is calculated as:

• \( \text{Period} = \frac{2\pi}{3} \)

This implies that the wave repeats its pattern every \( \frac{2\pi}{3} \) units on the horizontal axis.

• The smaller the value of B, the longer the period.
• A larger period means the wave stretches out along the x-axis.

Conversely, a higher value of B leads to a shorter period, causing the wave to condense and repeat more frequently.

Graphing Trigonometric Functions

Graphing trigonometric functions like the sine function involves a few simple steps. It's all about plotting points based on the function's characteristics and then drawing the wave.

The sine wave, known for its characteristic curves, looks like a rolling ocean wave and repeats at regular intervals.

• Start by identifying key points: the zero crossings, peaks, and troughs.
• For \(y = 4 \sin 3x\), consider:
  - The starting point at (0,0)
  - The peak at \(\left(\frac{\pi}{6}, 4\right)\)
  - Crossing again at \(\left(\frac{\pi}{3}, 0\right)\)
  - The trough at \(\left(\frac{\pi}{2}, -4\right)\)

Connect these points smoothly, ensuring the wave's shape reflects the sinusoidal nature, with crests reaching the amplitude and how it repeats in a period of \(\frac{2\pi}{3}\). Graphing makes the mathematical function visible and understandable, facilitating the concept of oscillation and periodicity. Each element reveals the dynamic behavior of the sine function across its domain.