Finding The Y-intercept Of Y Equals -1/2 Cos(3/2 X)

The $y$-intercept is a fundamental concept in mathematics, especially when dealing with functions and their graphical representations. Understanding how to find the $y$-intercept is crucial for analyzing and interpreting various mathematical models. In this comprehensive guide, we will delve into the process of determining the $y$-intercept of the function $y = -\frac{1}{2} \cos(\frac{3}{2}x)$, breaking down each step to ensure clarity and a solid understanding. We will not only solve the problem at hand but also explore the underlying principles and concepts, making this a valuable resource for anyone studying trigonometric functions.

Decoding the $y$-intercept

Before we dive into the specifics of the given function, let's first establish a clear understanding of what the $y$-intercept represents. The $y$-intercept is the point where the graph of a function intersects the $y$-axis. In simpler terms, it's the $y$-coordinate of the point where the graph crosses the vertical axis. This point is characterized by having an $x$-coordinate of 0. Therefore, to find the $y$-intercept of any function, we set $x = 0$ and solve for $y$. This seemingly simple concept is the key to unlocking a wealth of information about a function's behavior and its graphical representation.

The $y$-intercept provides valuable insight into the function's behavior at the point where the input is zero. It helps us visualize the starting point of the function on the coordinate plane and can be a critical piece of information when modeling real-world phenomena. In many applications, the $y$-intercept represents an initial condition or a baseline value. For instance, in a graph representing the growth of a population, the $y$-intercept might represent the initial population size. Similarly, in a graph depicting the decay of a radioactive substance, the $y$-intercept could represent the initial amount of the substance. Therefore, mastering the technique of finding the $y$-intercept is not just a mathematical exercise but a practical skill with far-reaching implications.

Navigating the Cosine Function

The function we are dealing with, $y = -\frac{1}{2} \cos(\frac{3}{2}x)$, involves the cosine function, a cornerstone of trigonometry. The cosine function, denoted as $\cos(x)$, is a periodic function that oscillates between -1 and 1. Its graph is a wave-like curve that repeats itself every $2\pi$ radians. Understanding the basic properties of the cosine function is essential for determining the $y$-intercept of our given function. The standard cosine function, $y = \cos(x)$, has a $y$-intercept at the point (0, 1). This is because the cosine of 0 radians is equal to 1. However, our function is a modified version of the standard cosine function, with a vertical compression and a horizontal scaling, which will affect the $y$-intercept.

The vertical compression is introduced by the coefficient $-\frac{1}{2}$ in front of the cosine function. This coefficient scales the output of the cosine function by a factor of $-\frac{1}{2}$. In other words, it compresses the graph vertically and reflects it across the $x$-axis. The horizontal scaling is introduced by the factor $\frac{3}{2}$ inside the cosine function. This factor affects the period of the cosine function, which is the length of one complete cycle. The period of the standard cosine function is $2\pi$, but the period of our function is given by $\frac{2\pi}{\frac{3}{2}} = \frac{4\pi}{3}$. While the horizontal scaling affects the period and the $x$-intercepts, it does not directly influence the $y$-intercept. The $y$-intercept is solely determined by the vertical scaling and the value of the cosine function at $x = 0$.

The $y$-intercept Calculation

Now that we have a firm grasp of the underlying concepts, let's proceed with the calculation of the $y$-intercept for the function $y = -\frac{1}{2} \cos(\frac{3}{2}x)$. As we discussed earlier, to find the $y$-intercept, we need to set $x = 0$ and solve for $y$. Substituting $x = 0$ into the function, we get:

$y = -\frac{1}{2} \cos(\frac{3}{2} \cdot 0)$

Simplifying the expression inside the cosine function:

$y = -\frac{1}{2} \cos(0)$

We know that the cosine of 0 radians is 1, so:

$y = -\frac{1}{2} \cdot 1$

Finally, we get:

$y = -\frac{1}{2}$

Therefore, the $y$-intercept of the function $y = -\frac{1}{2} \cos(\frac{3}{2}x)$ is $-\frac{1}{2}$. This means that the graph of the function intersects the $y$-axis at the point $(0, -\frac{1}{2})$. This point represents the value of the function when the input $x$ is zero. In the context of a graph, it's the point where the curve crosses the vertical axis. This simple calculation is a powerful tool for understanding the behavior of the function and visualizing its graph.

Selecting the Correct Option

Having calculated the $y$-intercept, we can now confidently identify the correct option from the given choices. The options provided are:

A. $\left(-\frac{1}{2}, 0\right)$ B. $\left(0, -\frac{1}{2}\right)$ C. $\left(0, \frac{3}{2}\right)$ D. $\left(\frac{3}{2}, 0\right)$

Recall that the $y$-intercept is a point on the coordinate plane, represented as $(x, y)$. We found that the $y$-intercept occurs when $x = 0$ and $y = -\frac{1}{2}$. Therefore, the correct point representing the $y$-intercept is $(0, -\frac{1}{2})$. Comparing this with the given options, we can see that option B, $\left(0, -\frac{1}{2}\right)$, matches our calculated $y$-intercept. The other options represent different points on the coordinate plane and do not correspond to the $y$-intercept of the given function. Option A, $\left(-\frac{1}{2}, 0\right)$, represents the $x$-intercept, which is the point where the graph intersects the $x$-axis. Options C and D do not have any direct significance in the context of the $y$-intercept.

Therefore, the correct answer is B. $\left(0, -\frac{1}{2}\right)$. This confirms that our calculations and understanding of the $y$-intercept are accurate. The ability to correctly identify the $y$-intercept is a crucial skill in mathematics, allowing for a deeper understanding of functions and their graphical representations.

Expanding the Horizon: Beyond the $y$-intercept

While finding the $y$-intercept is a significant step in understanding a function, it's just one piece of the puzzle. To gain a comprehensive understanding of a function, we need to explore other key features, such as the $x$-intercepts, the period, the amplitude, and the phase shift. The $x$-intercepts are the points where the graph intersects the $x$-axis, and they are found by setting $y = 0$ and solving for $x$. The period of a trigonometric function is the length of one complete cycle, and it determines how often the function repeats itself. The amplitude is the vertical distance between the midline and the maximum or minimum value of the function. The phase shift is a horizontal shift of the graph, which can be introduced by adding or subtracting a constant from the input variable.

In the case of the function $y = -\frac{1}{2} \cos(\frac{3}{2}x)$, we have already found the $y$-intercept. To find the $x$-intercepts, we would set $y = 0$ and solve for $x$. The period of the function is $\frac{4\pi}{3}$, as we discussed earlier. The amplitude of the function is |-rac{1}{2}| = \frac{1}{2}, which represents the maximum deviation of the function from the $x$-axis. There is no phase shift in this function, as there is no constant added or subtracted from the input variable $x$. By analyzing these features in addition to the $y$-intercept, we can create a detailed sketch of the graph and gain a thorough understanding of the function's behavior. This holistic approach to function analysis is essential for solving complex mathematical problems and applying mathematical concepts to real-world scenarios.

Conclusion

In this comprehensive guide, we have successfully determined the $y$-intercept of the function $y = -\frac{1}{2} \cos(\frac{3}{2}x)$. We began by establishing a clear understanding of the $y$-intercept and its significance in the context of functions and their graphs. We then delved into the properties of the cosine function and how transformations such as vertical compression and horizontal scaling affect its behavior. By setting $x = 0$ and solving for $y$, we calculated the $y$-intercept to be $-\frac{1}{2}$, which corresponds to the point $(0, -\frac{1}{2})$. This allowed us to confidently select the correct option from the given choices. Furthermore, we expanded our discussion to encompass other key features of functions, such as $x$-intercepts, period, amplitude, and phase shift, emphasizing the importance of a holistic approach to function analysis. Mastering the technique of finding the $y$-intercept, along with these other features, is a crucial step in developing a strong foundation in mathematics and its applications.