Problem 6 - Algebra and Trigonometry

Question

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=-4 \sin x$$

Step-by-Step Solution

Answer

The amplitude is 4. The function \( y = -4 \sin x \) is the graph of sine(x) reflected through the x-axis, and magnified vertically by a factor of 4.

Step 1: Find the amplitude

The amplitude of the function \( y = -4 \sin x \) is given by the absolute value of the coefficient of the \( \sin x \). That is, \( |-4| = 4 \). Therefore, the amplitude of \( y = -4 \sin x \) is 4.

Step 2: Plotting \( y = \sin x \)

Now draw the graph of \( y = \sin x \) for \( 0 \leq x \leq 2 \pi \). This is a wave that starts at 0, goes up to 1 at \( \frac{\pi}{2} \), down to 0 at \( \pi \), down to -1 at \( \frac{3 \pi}{2} \), and then back up to 0 at \( 2\pi \).

Step 3: Plotting \( y = -4 \sin x \)

Similarly, for the function \( y = -4 \sin x \), the graph will start at 0, but it will drop down to -4 at \( \frac{\pi}{2} \), go up to 0 at \( \pi \), up to 4 at \( \frac{3 \pi}{2} \), and then back down to 0 at \( 2\pi \). This is basically the same wave as \( y = \sin x \), but it's been flipped over the x-axis and stretched vertically by a factor of 4.

Key Concepts

Sine Function

Understanding the sine function is crucial in trigonometry. It is a periodic function that describes a smooth, wave-like pattern. The standard sine function is expressed as \( y = \sin x \). It has a period of \( 2\pi \) and an amplitude of 1, meaning it oscillates between -1 and 1.

The sine function is used to model phenomena such as sound waves, light waves, and tides, essentially any periodic behavior. It starts at the origin (0,0), rises to 1 at \( x = \frac{\pi}{2} \), descends back to 0 at \( x = \pi \), drops to -1 at \( x = \frac{3\pi}{2} \), and returns to 0 at \( x = 2\pi \).

This function is a base component in the graph of trigonometric functions, forming the backbone for transformations and variations in trigonometry.

Graph of Trigonometric Functions

Graphing trigonometric functions helps visualize how they behave over a specific interval. The sine function has a characteristic wave shape with a consistent pattern.

When plotting \( y = \sin x \) over the interval \( 0 \leq x \leq 2\pi \), it traces a single wave cycle. Key points to mark include:

• Starting at \( (0,0) \)
• Peaking at \( (\frac{\pi}{2}, 1) \)
• Crossing back through zero at \( (\pi,0) \)
• Valley at \( (\frac{3\pi}{2}, -1) \)
• Ending the cycle at \( (2\pi,0) \)

Graphing enables one to observe the symmetry about the origin or midline and the amplitude, which is the height from the centerline to the peak or trough. For altered functions like \( y = -4 \sin x \), this visual interpretation becomes even more important to understand the transformations that adjust the basic sine wave.

Transformations of Sine Function

Transformations modify the standard sine function to fit specific situations. This can involve changing its amplitude, period, and phase shift. Amplitude refers to the height of the wave. In the function \( y = -4 \sin x \), the amplitude is 4, larger than the standard 1, indicating the wave's peaks are four times higher (or lower) than usual.

The negative sign in \( y = -4 \sin x \) reflects the graph over the x-axis. This flipping inverts the wave, so it initially goes downward instead of upward. The function also involves vertical stretching because the peaks and troughs are farther from the axis.

Understanding these transformations is key, especially for applications requiring adjusted periodic models. This alters not just visual appearance but also the mathematical implications of real-world modeling.