Problem 113 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using the equation \(y=A \sin B x,\) if 1 replace either \(A\) or \(B\) with its opposite, the graph of the resulting equation is a reflection of the graph of the original equation about the \(x\) -axis.

Step-by-Step Solution

Answer

The statement 'Using the equation \(y=A \sin Bx\), if you replace either \(A\) or \(B\) with its opposite, the graph of the resulting equation is a reflection of the graph of the original equation about the \(x\)-axis' doesn't make sense entirely. Replacing A with -A results in a reflection about the \(x\)-axis. However, replacing B with -B results in a reflection about the \(y\)-axis, not the \(x\)-axis.

Step 1: Understanding the Underlying Sine Function

Observe the standard sine function \(y=\sin x\). Its graph oscillates above and below the \(x\)-axis, with peaks at \(y=1\) and troughs at \(y=-1\). The graph is symmetrical about the origin.

Step 2: Impact of Reversing A

Consider the case when A is replaced with -A, we get \(y=-A \sin Bx\). The graph of this function also oscillates above and below the \(x\)-axis, but the peaks and troughs are reversed - peaks are at \(y=-1\) and troughs are at \(y=1\). This graph is a reflection of the original function about the \(x\)-axis.

Step 3: Impact of Reversing B

Now, consider the case when B is replaced with -B, we get \(y=A \sin -Bx\). In this case, the graph of the function doesn't reflect about the \(x\)-axis, but rather it reflects about the \(y\)-axis. This is because changing the sign of B effectively changes the direction of the oscillations.

Key Concepts

Sine Function

The sine function is one of the foundational trigonometric functions represented by the equation \(y = \sin x\). It is essentially a wave that oscillates between -1 and 1, creating a periodic and smooth curve. This oscillation means that the graph frequently crosses the x-axis.

• The highest points, or peaks, of the sine wave reach up to \(y = 1\).
• The lowest points, or troughs, go down to \(y = -1\).
• It is symmetric around the origin, meaning it reflects evenly on either side of the y-axis.

The function's behavior is fundamental in many applications, like modeling sound waves or light waves, because of this regular, repeating pattern. The sine function is periodic with a period of \(2\pi\), meaning its shape repeats every \(2\pi\) units along the x-axis.

Graph Reflections

Reflections in graphs concern how a graph can flip over a particular axis. For the sine function given by \(y = A \sin Bx\):

• Switching the sign of \(A\) to \(-A\) flips the graph over the x-axis. High points become low and low points become high. For instance, if a point on the graph was at \((x, y)\), it becomes \((x, -y)\).
• Changing \(B\) to \(-B\), however, results in a reflection over the y-axis. This means the direction of the wave's travel is reversed, effectively flipping the graph horizontally without affecting its peaks and troughs.

These changes illustrate how reflections affect sine waves differently based on whether you alter the amplitude (peak height with \(A\)) or the frequency (oscillation rate with \(B\)). Understanding these reflections is crucial for manipulating graphs in various math and physics applications.

Amplitude and Frequency

The concepts of amplitude and frequency are crucial to understanding the behavior of the sine function described by \(y = A \sin Bx\).

• Amplitude: This is determined by the coefficient \(A\). It dictates the vertical stretch or compression of the graph. The amplitude is the distance from the middle of the wave (the x-axis in this case) to the peak or trough. If \(A\) is negative, it additionally flips the graph over the x-axis.
• Frequency: This is controlled by the variable \(B\). It affects how rapidly the sine function oscillates as x changes. A higher \(B\) results in more oscillations in a given interval, indicating a higher frequency. If \(B\) is negative, it doesn't affect the frequency magnitude, but it reflects the graph over the y-axis.

Both amplitude and frequency play vital roles in tailoring the sine function for specific needs, whether in simple demonstrations of periodic behavior or in more complex scenarios like signal processing.