Analyzing the Transformation
Our main focus is on the function $m(x) = -\sin(\frac{1}{4}x)$. To decipher the transformations, we need to compare it to the parent sine function, which is $f(x) = \sin(x)$. By examining the differences in the equation, we can identify the specific transformations that have occurred. These transformations affect the shape and position of the sine wave, influencing its key characteristics.
1. Frequency and Period: Let's dive into frequency and period. Frequency, guys, tells us how many cycles of the sine wave occur within a given interval, usually $2\pi$. The period, on the other hand, is the length of one complete cycle. These two are inversely related – a higher frequency means a shorter period, and vice versa. In our transformed function, the coefficient inside the sine function, which is $\frac{1}{4}$, plays a pivotal role in determining the frequency and period. This coefficient affects how "stretched" or "compressed" the sine wave is horizontally.
To calculate the period of the transformed function, we use the formula: $Period = \frac{2\pi}{|B|}$, where $B$ is the coefficient of $x$ inside the sine function. In our case, $B = \frac{1}{4}$, so the period is $Period = \frac{2\pi}{|\frac{1}{4}|} = 8\pi$. This means one complete cycle of the function $m(x)$ takes $8\pi$ units, which is four times longer than the period of the parent sine function (which is $2\pi$). Since the period has increased by a factor of 4, the frequency, which is the inverse of the period, has decreased by a factor of 4. Think of it like this: if the wave is stretched out horizontally, it will oscillate less frequently.
2. Amplitude and Reflection: Now, let's talk about amplitude and reflections. The amplitude of a sine function is the distance from the midline (the horizontal axis in the parent sine function) to the peak or trough of the wave. It essentially measures the "height" of the wave. The coefficient outside the sine function determines the amplitude. In our function, $m(x) = -\sin(\frac{1}{4}x)$, the coefficient is -1. The absolute value of this coefficient gives us the amplitude, which is $|-1| = 1$. So, the amplitude of $m(x)$ is the same as the parent sine function, which also has an amplitude of 1. This means the wave's vertical stretch or compression remains unchanged.
However, the negative sign in front of the sine function is crucial. It indicates a reflection across the x-axis. Imagine flipping the parent sine wave upside down – that's what the negative sign does. The peaks become troughs, and the troughs become peaks. So, while the amplitude remains the same, the reflection significantly alters the graph's appearance.
Determining the True Statements
Now that we've analyzed the transformations, let's evaluate the statements about the function $m(x) = -\sin(\frac{1}{4}x)$. Remember, we found that the period increased by a factor of 4, the frequency decreased by a factor of 4, the amplitude remained the same, and there was a reflection across the x-axis. With these key insights, we can confidently determine which statements accurately describe the transformation.
Let's revisit the provided statement options, keeping our analysis in mind. We'll carefully consider each option in light of our findings about frequency, period, amplitude, and reflections. By systematically evaluating each statement, we can pinpoint the ones that hold true for the transformation from the parent sine function to $m(x)$.
Given the transformation $m(x) = -\sin(\frac{1}{4}x)$, let's analyze the truthfulness of the following statements:
A. The frequency increases by a factor
Based on our previous discussion, we know that the frequency is related to the coefficient of $x$ inside the sine function. In this case, the coefficient is $\frac{1}{4}$. The general formula for the period of a transformed sine function is $Period = \frac{2\pi}{|B|}$, where $B$ is the coefficient of $x$. Therefore, the period of $m(x)$ is $Period = \frac{2\pi}{|\frac{1}{4}|} = 8\pi$.
The period of the parent sine function, $f(x) = \sin(x)$, is $2\pi$. Comparing the periods, we see that the period of $m(x)$ is four times the period of the parent sine function. Since frequency is the inverse of the period, if the period increases by a factor of 4, the frequency decreases by a factor of 4. Therefore, statement A is incorrect. The frequency decreases, not increases.
Conclusion
Understanding transformations of trigonometric functions involves carefully analyzing the coefficients and signs within the function's equation. By comparing the transformed function to the parent function, we can identify changes in period, frequency, amplitude, and reflections. These transformations alter the shape and position of the graph, providing a deeper understanding of the function's behavior. Through systematic analysis and application of relevant formulas, we can accurately determine the true statements about these transformations.