Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=-\sin x$$

Step-by-Step Solution

Answer

Amplitude is 1. Negative sign inverts the standard sine graph.

Step 1: Identify the Function Type

The given function is \( y = -\sin x \). This is a sine function with a reflection over the x-axis due to the negative sign.

Step 2: Write Down the Function's Amplitude Formula

For a sine function of the form \( y = A \sin(Bx + C) + D \), the amplitude is the absolute value of \( A \). In this case, \( A = -1 \).

Step 3: Calculate the Amplitude

Since the amplitude is the absolute value of \( A \), we have \( \text{Amplitude} = |-1| = 1 \).

Step 4: Describe the Graphing Interval

The graph should be plotted over the interval \([-2\pi, 2\pi]\). This interval includes two complete periods of the sine function since the period of \( \sin x \) is \( 2\pi \).

Step 5: Consider the Effect of Reflection

The negative sign in the function \(-\sin x\) reflects the graph of \( \sin x \) across the x-axis. Peaks become troughs and vice-versa.

Step 6: Sketch the Graph

To graph \( y = -\sin x \), begin by plotting key points where \( x = {-2\pi, -\pi, 0, \pi, 2\pi} \). At these points, the values of \( y \) are 0, 0, 0, 0, and 0 respectively, since sine is 0 at these multiples of \( \pi \). Between these points, the graph reaches troughs at \( x = -3\pi/2 \) and \( x = -\pi/2 \), as well as \( x = \pi/2 \) and \( x = 3\pi/2 \).

Key Concepts

Sine Function

The sine function is one of the most fundamental trigonometric functions, often represented as \( y = \sin x \). This function is periodic with a period of \( 2\pi \), meaning the pattern repeats itself every \( 2\pi \) units along the x-axis. Sine functions are essential in modeling wave-like phenomena such as sound waves, light waves, and tides.

• The function oscillates between -1 and 1, creating a smooth wave-like shape that starts at 0, rises to 1, falls to -1, and returns to 0 over its period.
• This wave-like nature is a reason sine functions are so useful in physics and engineering.
• Every sine graph has intercepts where it crosses the x-axis, typically located at multiples of \( \pi \), such as 0, \( \pi \), \( -\pi \), and so on.

In trigonometry, the argument \( x \) can be an angle measured in radians or degrees, both of which influence the shape and periodicity of the curve.

Amplitude

Amplitude refers to the height of the wave from the center line (equilibrium) to its peak or trough. In a sine function of the form \( y = A \sin(Bx + C) + D \), the amplitude is given by the absolute value of \( A \). It determines how "tall" or "short" the curve appears.

• For the standard sine function, \( y = \sin x \), the amplitude is 1, as it reaches a maximum of 1 and a minimum of -1.
• In the case of the given function \( y = -\sin x \), the amplitude is \( |-1| = 1 \). The negative sign affects direction but not amplitude magnitude.
• The amplitude showcases how vigorous the oscillation of the sine wave is, which can be crucial in applications like signal processing.

Thus, even if the wave is reflected or shifted, the amplitude helps us understand the level of intensity relative to the original function.

Reflection

Reflection in trigonometry refers to flipping or mirroring a graph across a specific axis. For sinusoidal functions, like the sine function, an important transformation is reflection over the x-axis.

• In the expression \( y = -\sin x \), the negative sign before the sine flips the graph upside down.
• This means that maxima (peaks) of \( \sin x \) become minima (troughs) for \( -\sin x \) and vice versa.
• Reflection is crucial in understanding how waves behave in different physical contexts, such as sound bouncing back when hitting a wall or light reflecting off a surface.

This graphical transformation does not change other properties of the function like period or amplitude, but it significantly impacts the visual representation and interpretation of data or phenomena.