Problem 35 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function \( y = -\sin 2x \) completes one cycle from 0 to \( \pi \), starting and ending at 0, with a minimum at \( \pi/4 \) and a maximum at \( 3\pi/4 \).

Step 1: Understand the Standard Sine Wave

The standard sine wave function is given by \( y = \sin x \). One complete cycle of \( y = \sin x \) occurs from \( 0 \) to \( 2\pi \) radians. In this interval, the sine function starts at 0, goes to a maximum of 1 at \( \pi/2 \), back to 0 at \( \pi \), down to a minimum of -1 at \( 3\pi/2 \), and finally returns to 0 at \( 2\pi \).

Step 2: Consider the Effects of the Coefficient 2

For our function \( y = -\sin 2x \), the coefficient 2 inside the function alters the period of the sine wave. The period of a sine wave with equation \( y = \sin(bx) \) is given by \( \frac{2\pi}{b} \). So for \( b = 2 \), the period is \( \frac{2\pi}{2} = \pi \). This means one cycle of \( y = -\sin 2x \) will repeat every \( \pi \) radians.

Step 3: Reflect the Function

The negative sign in \( y = -\sin 2x \) reflects the graph of \( y = \sin 2x \) across the x-axis. This means that all positive values of \( \sin 2x \) become negative and vice versa.

Step 4: Sketch the Graph Over One Cycle

To sketch \( y = -\sin 2x \), note that in one cycle (from 0 to \( \pi \)), the function will start at 0, move to a minimum of -1 at \( \pi/4 \), return to 0 at \( \pi/2 \), reach a maximum of 1 at \( 3\pi/4 \), and return to 0 at \( \pi \).

Step 5: Plot the Key Points and Draw the Curve

Based on the above analysis, plot the key points: (0,0), (\( \pi/4,-1 \)), (\( \pi/2,0 \)), (\( 3\pi/4,1 \)), and (\( \pi,0 \)). Connect these points with a smooth, sinusoidal curve to complete the sketch of one cycle.

Key Concepts

Sine Function

The sine function is a foundational trigonometric function often represented by the equation \( y = \sin x \). It is a periodic function, which means it repeats its values in regular intervals. Sine functions are incredibly useful in modeling wave-like phenomena, such as sound and light waves.

When graphed, the sine wave forms a smooth, continuous curve that oscillates above and below the x-axis. Here's how it behaves in one \("standard cycle"\):

• Starts at 0
• Rises to a maximum of 1 at \( \pi/2 \)
• Returns to 0 at \( \pi \)
• Dips to a minimum of -1 at \( 3\pi/2 \)
• Completes the cycle back at 0 by \( 2\pi \)

These cyclic variations are what make it easy to model repeating patterns in nature. Understanding its basic properties helps in sketching transformations of the sine wave, like in our original problem.

Period of a Function

The period of a function in mathematics tells us the length of one complete cycle of a periodic function. For the standard sine function, \( y = \sin x \), the period is \( 2\pi \). This means that every \( 2\pi \) units, the sine function starts repeating itself.

In a modified sine function like \( y = \sin(bx) \), the coefficient \( b \) affects the period. The formula \( \frac{2\pi}{b} \) is used to calculate the new period. Applying this to our example \( y = -\sin 2x \), we set \( b \) to 2, which results in a shorter period of \( \pi \).

• The function now completes one cycle in just \( \pi \) radians.
• This doubling in frequency makes the wave oscillate more quickly.

Understanding how the period changes with different coefficients is essential when sketching and interpreting trigonometric functions. It's key to replicating the behavior of phenomena that have their own rhythmic patterns.

Graph Reflection

Graph reflection refers to the flipping of a graph across an axis. In trigonometric graphs, such as those of sine and cosine, reflections can help depict a variety of waveforms and signals. For the function \( y = -\sin x \), the graph of the original function \( y = \sin x \) is reflected over the x-axis.

This reflection affects every point on the sine wave:

• Positive peaks turn into negative valleys.
• Negative valleys turn into positive peaks.
• Zeros remain unchanged, maintaining their positions at the x-axis intercepts.

For our specific function, \( y = -\sin 2x \), the reflection means altering not only the amplitude but also shifting the phase of the sine wave. Each peak and trough swaps sides of the x-axis, creating a mirror image of the graph. This concept of reflection helps us understand real-world applications where signals might invert, such as electrical signals seen on oscilloscopes. Reflecting graphs also helps in visualizing and solving trigonometric equations that involve negative signs.