Finding The Minimum Value Of Y=-3+4 Cos(5π/6(x+4))

In the realm of mathematics, trigonometric functions play a pivotal role in modeling periodic phenomena. Understanding the behavior of these functions, including their maximum and minimum values, is crucial in various applications, from physics to engineering. This article delves into the process of determining the minimum value of a trigonometric function, specifically focusing on the function $y = -3 + 4 \cos(\frac{5\pi}{6}(x+4))$.

Understanding the Cosine Function

To effectively find the minimum value, we must first grasp the fundamental properties of the cosine function. The cosine function, denoted as cos(x), is a periodic function that oscillates between -1 and 1. Its graph exhibits a wave-like pattern, with the highest point (maximum) at 1 and the lowest point (minimum) at -1. This inherent characteristic of the cosine function is the cornerstone for determining the minimum value of more complex trigonometric expressions.

The cosine function's range is [-1, 1]. This means that for any real number x, the value of cos(x) will always be between -1 and 1, inclusive. Understanding this range is essential for finding the minimum and maximum values of trigonometric functions that involve cosine. The basic cosine function, y = cos(x), has a maximum value of 1 (occurring at x = 2nπ, where n is an integer) and a minimum value of -1 (occurring at x = (2n+1)π, where n is an integer).

When we deal with transformations of the cosine function, such as amplitude changes, vertical shifts, horizontal shifts, and period changes, we need to consider how these transformations affect the range. For instance, a vertical shift upwards will increase both the maximum and minimum values, while an amplitude change will stretch or compress the range. A negative sign in front of the cosine function reflects the graph across the x-axis, swapping the positions of the maximum and minimum values. Therefore, a solid grasp of the cosine function's range and how it transforms is crucial for accurately determining minimum and maximum values.

Analyzing the Given Function

Now, let's turn our attention to the function at hand: $y = -3 + 4 \cos(\frac{5\pi}{6}(x+4))$. This function is a transformation of the basic cosine function, and we can identify several key components:

• Amplitude: The amplitude is the absolute value of the coefficient of the cosine function, which in this case is |4| = 4. The amplitude determines the vertical stretch of the function. This means that the cosine function's typical range of [-1, 1] is stretched to [-4, 4]. The amplitude dictates how far the function deviates from its midline.
• Vertical Shift: The constant term added to the cosine function is -3. This represents a vertical shift of the entire graph downward by 3 units. The vertical shift affects the midline of the function. In this case, the midline shifts from y = 0 to y = -3.
• Horizontal Shift: The term (x + 4) inside the cosine function indicates a horizontal shift. Specifically, it shifts the graph 4 units to the left. This horizontal shift does not affect the minimum or maximum values, but it changes the x-values at which these extrema occur.
• Period: The term $\frac{5\pi}{6}$ multiplied by (x + 4) affects the period of the function. The period is the horizontal distance it takes for the function to complete one full cycle. The period is calculated as $\frac{2\pi}{B}$, where B is the coefficient of x. In this case, B = $\frac{5\pi}{6}$, so the period is $\frac{2\pi}{\frac{5\pi}{6}} = \frac{12}{5}$. The period influences how frequently the function oscillates, but like the horizontal shift, it does not alter the extreme values.

Determining the Minimum Value

To find the minimum value of the function, we need to consider how each of these transformations affects the cosine function's inherent range. We know that the cosine function itself has a minimum value of -1. The function $y = -3 + 4 \cos(\frac{5\pi}{6}(x+4))$ can be analyzed step by step:

1. The cosine part, $\cos(\frac{5\pi}{6}(x+4))$, has a minimum value of -1.
2. Multiplying by 4 gives $4 \cos(\frac{5\pi}{6}(x+4))$, which has a minimum value of 4 * (-1) = -4.
3. Adding -3 to the entire expression shifts the minimum value downward by 3 units. Therefore, the minimum value of the function is -4 + (-3) = -7.

Thus, the minimum value of the function $y = -3 + 4 \cos(\frac{5\pi}{6}(x+4))$ is -7. This value occurs when $\cos(\frac{5\pi}{6}(x+4)) = -1$. To find the x-values at which this minimum occurs, we set $\frac{5\pi}{6}(x+4) = (2n+1)\pi$, where n is an integer. Solving for x gives $x = \frac{6}{5}(2n+1) - 4$.

In summary, by understanding the transformations applied to the basic cosine function, we can systematically determine the minimum value of the given function. The amplitude stretches the range, the vertical shift moves the midline, and the minimum value is found by considering the combined effect of these transformations.

Graphical Representation

Visualizing the function graphically can provide a deeper understanding of its behavior. The graph of $y = -3 + 4 \cos(\frac{5\pi}{6}(x+4))$ is a cosine wave that has been stretched vertically by a factor of 4, shifted downward by 3 units, and shifted horizontally to the left by 4 units. The minimum value of -7 can be clearly seen as the lowest point on the graph.

The graph oscillates between -7 and 1, which are the minimum and maximum values, respectively. The midline of the graph is at y = -3, and the function completes one full cycle over a period of $\frac{12}{5}$ units. The horizontal shift affects the starting point of the cycle, but it does not change the vertical range or the extreme values.

Applications and Significance

The ability to determine the minimum and maximum values of trigonometric functions has numerous applications in various fields. In physics, these functions are used to model oscillations and waves, such as sound waves and electromagnetic waves. The minimum and maximum values correspond to the minimum and maximum amplitudes of these waves, which can represent physical quantities like pressure or voltage.

In engineering, trigonometric functions are used in the design of electrical circuits, mechanical systems, and signal processing algorithms. Understanding the range of these functions is crucial for ensuring that systems operate within acceptable limits and for optimizing their performance. For instance, in signal processing, the minimum and maximum values of a signal can determine its dynamic range and signal-to-noise ratio.

Furthermore, in mathematics itself, the concepts of minimum and maximum values are fundamental to calculus and optimization problems. Finding the extreme values of functions is essential for solving real-world problems involving optimization, such as maximizing profit or minimizing cost.

Conclusion

In conclusion, finding the minimum value of the function $y = -3 + 4 \cos(\frac{5\pi}{6}(x+4))$ involves understanding the properties of the cosine function and the effects of various transformations. By recognizing the amplitude, vertical shift, horizontal shift, and period, we can systematically determine that the minimum value of the function is -7. This process highlights the importance of trigonometric functions in modeling periodic phenomena and the significance of their minimum and maximum values in various applications. Mastering these concepts is crucial for anyone working in mathematics, physics, engineering, or related fields.