Problem 29 - Algebra 2 and Trigonometry

Question

Sketch one cycle of each function. \(y=\sin 2 x\)

Step-by-Step Solution

Answer

One cycle of \( y=\sin 2x \) spans from 0 to \( \pi \), with key points at \(0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \) and \( \pi \).

Step 1: Understanding the Function

The function given is \( y = \sin 2x \), which is a transformation of the basic sine function \( y = \sin x \). The multiplier of the angle, 2x, affects the function's period.

Step 2: Determine the Period

The period of a sine function is calculated as \( \frac{2\pi}{k} \), where \( k \) is the coefficient of \( x \) in \( \sin kx \). For \( y = \sin 2x \), \( k = 2 \), so the period is \( \frac{2\pi}{2} = \pi \). This means one full cycle of \( y = \sin 2x \) occurs between 0 and \( \pi \).

Step 3: Identify Key Points

The key points for one cycle of the sine function are at the intervals of its period. For \( y = \sin 2x \) within one cycle, these points include: \((0, 0)\), where the function starts; \(\left(\frac{\pi}{4}, 1\right)\), the maximum; \(\left(\frac{\pi}{2}, 0\right)\), the midline crossing; \(\left(\frac{3\pi}{4}, -1\right)\), the minimum; and \(\left(\pi, 0\right)\), where the cycle completes.

Step 4: Sketch the Function

On a graph, the x-axis is marked from 0 to \( \pi \) to account for one period, with increments to highlight the key points \(0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \) and \( \pi \). Plot the points \((0, 0)\), \(\left(\frac{\pi}{4}, 1\right)\), \(\left(\frac{\pi}{2}, 0\right)\), \(\left(\frac{3\pi}{4}, -1\right)\), and \((\pi, 0)\). Draw a smooth curve through these points, resembling the standard sine wave, but compressed horizontally to fit within \( \pi \).

Step 5: Verify the Sketch

Ensure that the sketch makes one complete wave (from 0 to \(\pi\)). It should start at zero, reach a maximum at \(\frac{\pi}{4}\), pass through zero at \(\frac{\pi}{2}\), hit a minimum at \(\frac{3\pi}{4}\), and return to zero at \(\pi\), verifying the completion of one full cycle.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function, represented as \( y = \sin x \). It's a periodic function with a classic wave-like shape, oscillating smoothly above and below the x-axis. At its core, it calculates the y-coordinate of a point on the unit circle (a circle with a radius of one) as it moves around, corresponding to the angle \( x \) in radians.

Basic characteristics of the sine function include:

• *Amplitude*: The maximum distance from the midline of the wave, typically 1 for \( y = \sin x \).
• *Period*: The length it takes to complete one full cycle. For \( \sin x \), the period is \( 2\pi \).
• *Range*: Outputs values fluctuating between -1 and 1.
• *Midline*: The horizontal axis around which it oscillates, often the x-axis.

Understanding the sine function's fundamentals allows us to recognize how transformations, like changes in amplitude, shifts, and stretches, can alter its shape and behavior.

Period of a Function

The period of a trigonometric function refers to the distance on the x-axis required for the function to start repeating its behavior. This periodic repetition is a crucial part of functions like \( \sin x \).

To determine the period of \( y = \sin kx \), use the formula:

• Period = \( \frac{2\pi}{k} \)

where \( k \) is the coefficient of \( x \). This formula shows that a greater \( k \) value will compress the cycle length, making more cycles fit into a given span.

For instance, if \( k = 2 \) in \( y = \sin 2x \), the period is calculated as:

• Period = \( \frac{2\pi}{2} = \pi \)

This demonstrates that \( y = \sin 2x \) completes a full cycle in \( \pi \) units, half the typical period of \( \sin x \).

The concept of the period becomes vital for graphing, especially when identifying key points on the graph over one cycle.

Graphing Trigonometric Functions

Graphing trigonometric functions like the sine function involves plotting its values over its period. Let's take the example of \( y = \sin 2x \) to illustrate the graphing process.

First, set the x-axis interval for one complete period. For \( y = \sin 2x \), this is from 0 to \( \pi \). Break this interval into segments reflecting key cycle points, such as maximum, minimum, and zero crossings, giving a clear guide to where the function’s critical points lie.

Key points for \( y = \sin 2x \) between 0 and \( \pi \) include:

• (0, 0): The start at the x-axis
• \( \left(\frac{\pi}{4}, 1\right)\): The maximum point
• \( \left(\frac{\pi}{2}, 0\right)\): Crossing the x-axis
• \( \left(\frac{3\pi}{4}, -1\right)\): The minimum point
• (\( \pi, 0\)): End of the cycle

Placing these points on a graph, draw a smooth curve that connects them. This curve should resemble the standard sine wave but compressed horizontally, fitting within a shorter length.

Accurate graphing of trigonometric functions not only helps in visualizing the function's behavior but also serves in confirming the understanding of its mathematical transformation.