Problem 24 - Algebra and Trigonometry

Question

\(17-28\) . Find the amplitude and period of the function, and sketch its graph. $$ y=4 \sin (-2 x) $$

Step-by-Step Solution

Answer

Amplitude: 4; Period: \(\pi\). Reflect over x-axis.

Step 1: Identify the Amplitude

The amplitude of a sinusoidal function is the coefficient in front of the sine function. For the function \( y = 4 \sin(-2x) \), the amplitude is the absolute value of 4, which is \( 4 \).

Step 2: Determine the Period

The period of a sine function \( y = a \sin(bx) \) can be calculated using the formula \( \frac{2\pi}{|b|} \). Here, \( b = -2 \). Calculate the period: \( \frac{2\pi}{|-2|} = \frac{2\pi}{2} = \pi \).

Step 3: Describe the Reflection

Since the sine function has a negative factor \(-2\), the function is reflected over the x-axis. However, the amplitude remains positive, as the amplitude is always given by the absolute value.

Step 4: Sketch the Graph

To sketch the graph, plot a sinusoidal curve starting at the origin. Since the function is \( y = 4 \sin(-2x) \), reflected over the x-axis, you start by going downward. Complete one cycle spanning from \( x = 0 \) to \( x = \pi \), reaching -4 at \( \frac{\pi}{2} \), back to 0 at \( \pi \), and continue the pattern.

Key Concepts

Sinusoidal Function

A sinusoidal function describes a smooth, periodic oscillation similar to the motion of waves. This type of function takes the form:

• \( y = a \sin(bx + c) + d \)

Here, the sine function is multiplied by a coefficient "\( a \)", which determines the amplitude, or the height of the wave's peaks and troughs. The "\( b \)" is a frequency component, which helps decide how many cycles occur in a specific interval.

Understanding sinusoidal functions is crucial as they appear in various real-world contexts like sound waves, light waves, and the motion of pendulums. The graph of a sine or cosine function showcases these wavelike patterns. When solving problems involving sinusoidal functions, ensuring clarity is key—understanding how each parameter affects the wave can aid in both sketching and interpreting graphs effectively.

Graphing Sine Functions

Graphing sine functions involves plotting a wave that starts at the origin and continues to repeat its pattern based on its period. For the equation \( y = 4 \sin(-2x) \), there are a few key:

• The amplitude is \( 4 \), indicating that the wave reaches from \( -4 \) to \( 4 \).
• The period is given by \( \frac{2\pi}{|-b|} \), where \( b = -2 \), resulting in a period of \( \pi \).
• The negative sign in the argument indicates a reflection, but more on that in the next section.

To start graphing, begin by plotting points using the calculated amplitude and period. The sine wave starts at zero, reaches a maximum at one-quarter of the period, crosses zero at one-half, reaches a minimum at three-quarters, and returns to zero at the period’s end.

Such graphs help visualize how quickly or slowly things change with respect to time or another parameter, making it an essential tool in physics and engineering.

Reflection of Sine Function

In the function \( y = 4 \sin(-2x) \), there is a negative sign before the \( 2x \), indicating reflection. Reflection over the x-axis means flipping the wave upside down. Instead of the sine curve starting and going upwards, it starts and moves downward, reflecting each point of the typical sine curve across the x-axis.

Characteristics of Reflected Sine Curves

• A mirror image across the horizontal axis, maintaining its period and amplitude:
• The usually upward stretch at the start becomes a downward stretch.
• Peaks become troughs and troughs become peaks.

Despite this reflection, the amplitude remains positive because it represents a distance. Considering reflections allow us to grasp how modifications in equations affect the visual representation, providing an insightful glimpse into the interplay between algebraic expressions and graphical outputs.