Transformations Of Sine Functions G(x) = (1/3)sin(2x)

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Introduction to Sine Function Transformations

Hey guys! Let's dive into the fascinating world of sine function transformations. Understanding these transformations is crucial for anyone delving into trigonometry and calculus. The parent sine function, denoted as $f(x) = \sin(x)$, is the foundation upon which various transformations are applied. Transformations alter the graph of the sine function in predictable ways, changing its amplitude, period, phase shift, and vertical shift. In this comprehensive guide, we’ll break down each type of transformation, providing you with the tools to analyze and manipulate sine functions effectively. We'll focus on how these transformations affect the function's graph and equation, using examples to illustrate each concept clearly. Understanding these concepts thoroughly will not only help you ace your exams but also provide a solid base for more advanced mathematical topics. So, buckle up, and let's embark on this mathematical journey together, making sine function transformations a piece of cake!

The general form of a transformed sine function can be expressed as: $g(x) = A \sin(B(x - C)) + D$, where each parameter plays a specific role in altering the parent sine function. Here, A represents the amplitude, B affects the period, C denotes the phase shift (horizontal shift), and D indicates the vertical shift. Each of these parameters manipulates the sine wave in a unique manner, and understanding their individual effects is key to mastering transformations. By grasping how each parameter changes the sine function's graph, you can easily analyze complex sinusoidal equations and even create your own transformations to fit specific scenarios. We'll explore each of these parameters in detail, providing clear examples and visual aids to help you understand the mechanics behind these transformations. This knowledge is invaluable for various applications, from physics and engineering to music and signal processing, where sinusoidal functions are used extensively to model periodic phenomena.

Amplitude Transformations

The amplitude, represented by $|A|$, determines the vertical stretch or compression of the sine wave. Think of it as the height of the wave from its midline. If $|A| > 1$, the sine function is stretched vertically, making the wave taller. Conversely, if $0 < |A| < 1$, the sine function is compressed vertically, squashing the wave down. For example, in the function $g(x) = 2 \sin(x)$, the amplitude is 2, meaning the graph oscillates between -2 and 2, twice as tall as the parent sine function. On the other hand, in $g(x) = 0.5 \sin(x)$, the amplitude is 0.5, compressing the graph to oscillate between -0.5 and 0.5. Understanding the amplitude transformation is crucial because it directly affects the range of the function, influencing its maximum and minimum values. This concept is particularly important in real-world applications, such as sound waves where amplitude corresponds to loudness, or electromagnetic waves where it relates to intensity.

Period Transformations

The period of a sine function is the length of one complete cycle, and it's influenced by the parameter $B$. The formula to calculate the period is $T = \frac{2\pi}{|B|}$. If $|B| > 1$, the period decreases, compressing the graph horizontally, making the wave cycles more frequent. If $0 < |B| < 1$, the period increases, stretching the graph horizontally, spreading out the wave cycles. For instance, in the function $g(x) = \sin(2x)$, $B = 2$, so the period is $T = \frac{2\pi}{2} = \pi$, which means the wave completes one full cycle in half the time compared to the parent sine function. Conversely, in $g(x) = \sin(\frac{1}{2}x)$, $B = \frac{1}{2}$, and the period is $T = \frac{2\pi}{\frac{1}{2}} = 4\pi$, doubling the time for a complete cycle. Period transformations are vital in scenarios where frequency is a key factor, such as in musical notes where frequency determines pitch, or in electrical engineering where it affects the frequency of alternating current.

Phase Shift Transformations

The phase shift, represented by $C$, determines the horizontal shift of the sine function. A positive $C$ shifts the graph to the right, while a negative $C$ shifts it to the left. Think of it as sliding the entire sine wave along the x-axis. In the function $g(x) = \sin(x - \frac{\pi}{2})$, the phase shift is $ rac{\pi}{2}$ units to the right, meaning the graph starts its cycle later than the parent sine function. Conversely, in $g(x) = \sin(x + \frac{\pi}{2})$, the phase shift is $ rac{\pi}{2}$ units to the left, causing the graph to start its cycle earlier. Phase shifts are crucial in applications where the timing or synchronization of waves is important, such as in signal processing where a phase shift can correct signal timing, or in optics where it can affect interference patterns.

Vertical Shift Transformations

The vertical shift, indicated by $D$, moves the sine function up or down along the y-axis. A positive $D$ shifts the graph upwards, while a negative $D$ shifts it downwards. This transformation changes the midline of the sine wave. For example, in the function $g(x) = \sin(x) + 3$, the graph is shifted 3 units upwards, so the midline becomes $y = 3$. Similarly, in $g(x) = \sin(x) - 2$, the graph is shifted 2 units downwards, making the midline $y = -2$. Vertical shifts are important in modeling real-world phenomena where the average value of the sinusoidal function is not zero, such as in temperature variations throughout the day, where the average temperature serves as the midline.

Analyzing the Given Transformation: $g(x) = \frac{1}{3} \sin(2x)$

Now, let's analyze the given transformation: $g(x) = \frac{1}{3} \sin(2x)$. This function combines both amplitude and period transformations, providing a practical example of how these parameters work together. By breaking down this equation, we can see the individual effects of each transformation on the parent sine function. This exercise will help solidify your understanding of how to interpret transformed sine functions and predict their graphical behavior.

Amplitude Analysis

First, let's consider the amplitude. In the function $g(x) = \frac{1}{3} \sin(2x)$, the coefficient $ rac{1}{3}$ in front of the sine function represents the amplitude. Since $ rac{1}{3} < 1$, this indicates a vertical compression of the sine wave. The graph will oscillate between $-\frac{1}{3}$ and $\frac{1}{3}$, making it one-third as tall as the parent sine function. This compression means the wave will have a reduced height, resulting in a flatter appearance compared to the standard sine wave. Understanding this amplitude change is key to accurately sketching the graph of the transformed function.

Period Analysis

Next, let's examine the period. The value inside the sine function, $2x$, affects the period of the wave. Here, $B = 2$, so we use the formula $T = \frac{2\pi}{|B|}$ to find the new period. Substituting $B = 2$, we get $T = \frac{2\pi}{2} = \pi$. This means the period of the transformed function is $\pi$, which is half the period of the parent sine function (which has a period of $2\pi$). As a result, the graph is compressed horizontally, and the wave completes one full cycle in half the usual distance. This compression leads to the wave oscillating more frequently within the same interval compared to the parent sine function.

Combined Effect

Combining these two transformations, the graph of $g(x) = \frac{1}{3} \sin(2x)$ is a sine wave that is both compressed vertically (due to the amplitude of $\frac{1}{3}$) and compressed horizontally (due to the period of $\pi$). This results in a wave that is shorter and more frequent than the standard sine wave. By understanding how these transformations combine, you can accurately visualize and sketch the graph of the transformed function. This combined effect highlights the power of transformations in shaping trigonometric functions to fit specific mathematical models and real-world applications.

Step-by-Step Solution for Specific Values

To further illustrate the transformation, let's evaluate $g(x) = \frac{1}{3} \sin(2x)$ at specific points. This will give us concrete values to plot and visualize the transformed function. By calculating these points, we can see exactly how the transformations affect the function's output at different inputs. This step-by-step approach is a valuable tool for understanding and graphing transformed trigonometric functions.

Evaluating at $x = 0$

First, let's evaluate $g(x)$ at $x = 0$:

g(0)=13sin⁡(2⋅0)=13sin⁡(0)=13⋅0=0g(0) = \frac{1}{3} \sin(2 \cdot 0) = \frac{1}{3} \sin(0) = \frac{1}{3} \cdot 0 = 0

So, at $x = 0$, $g(x) = 0$. This point remains unchanged from the parent sine function, as the transformations do not shift the origin in this case.

Evaluating at $x = \frac{\pi}{4}$

Next, let's evaluate $g(x)$ at $x = \frac{\pi}{4}$:

g(π4)=13sin⁡(2⋅π4)=13sin⁡(π2)=13⋅1=13g\left(\frac{\pi}{4}\right) = \frac{1}{3} \sin\left(2 \cdot \frac{\pi}{4}\right) = \frac{1}{3} \sin\left(\frac{\pi}{2}\right) = \frac{1}{3} \cdot 1 = \frac{1}{3}

At $x = \frac{\pi}{4}$, $g(x) = \frac{1}{3}$. This point is at its maximum value, but compressed to one-third of the parent sine function's maximum.

Evaluating at $x = \frac{\pi}{2}$

Now, let's evaluate $g(x)$ at $x = \frac{\pi}{2}$:

g(π2)=13sin⁡(2⋅π2)=13sin⁡(π)=13⋅0=0g\left(\frac{\pi}{2}\right) = \frac{1}{3} \sin\left(2 \cdot \frac{\pi}{2}\right) = \frac{1}{3} \sin(\pi) = \frac{1}{3} \cdot 0 = 0

At $x = \frac{\pi}{2}$, $g(x) = 0$. This shows the function has completed half of its compressed cycle, returning to the midline.

Evaluating at $x = \frac{3\pi}{4}$

Let's evaluate $g(x)$ at $x = \frac{3\pi}{4}$:

g(3π4)=13sin⁡(2⋅3π4)=13sin⁡(3π2)=13⋅(−1)=−13g\left(\frac{3\pi}{4}\right) = \frac{1}{3} \sin\left(2 \cdot \frac{3\pi}{4}\right) = \frac{1}{3} \sin\left(\frac{3\pi}{2}\right) = \frac{1}{3} \cdot (-1) = -\frac{1}{3}

At $x = \frac{3\pi}{4}$, $g(x) = -\frac{1}{3}$. This is the minimum value of the transformed function, compressed to one-third of the parent sine function's minimum.

Evaluating at $x = \pi$

Finally, let's evaluate $g(x)$ at $x = \pi$:

g(π)=13sin⁡(2⋅π)=13sin⁡(2π)=13⋅0=0g(\pi) = \frac{1}{3} \sin(2 \cdot \pi) = \frac{1}{3} \sin(2\pi) = \frac{1}{3} \cdot 0 = 0

At $x = \pi$, $g(x) = 0$. This completes one full compressed cycle, returning the function to its starting point.

By plotting these points (0, 0), ($\frac{\pi}{4}$, $\frac{1}{3}$), ($\frac{\pi}{2}$, 0), ($\frac{3\pi}{4}$, -$\frac{1}{3}$), and ($\pi$, 0), you can clearly visualize the transformed sine function. The graph is compressed both vertically and horizontally, showcasing the combined effects of amplitude and period transformations.

Conclusion

In conclusion, mastering transformations of sine functions is a fundamental skill in mathematics, with applications ranging from physics to engineering. By understanding how amplitude, period, phase shift, and vertical shift affect the sine wave, you can analyze and manipulate these functions with confidence. Remember, the general form $g(x) = A \sin(B(x - C)) + D$ is your key to unlocking the behavior of any transformed sine function. Keep practicing, and soon you'll be transforming sine functions like a pro! Understanding these concepts not only enhances your mathematical toolkit but also provides valuable insights into the world around us, where sinusoidal patterns are prevalent in various phenomena. So, keep exploring and transforming, guys! You've got this!

This comprehensive guide has equipped you with the knowledge to tackle various transformations of sine functions. From amplitude and period changes to phase shifts and vertical shifts, you now have a solid understanding of how these parameters shape the sine wave. By breaking down the function $g(x) = \frac{1}{3} \sin(2x)$, we've demonstrated how to analyze and visualize transformed functions step by step. Keep practicing, and you'll become proficient in manipulating trigonometric functions, opening doors to a deeper understanding of mathematical modeling and its applications in real-world scenarios.