Анализ И График Функции: Y = 2sin(x - Π/6) - 2

Hey guys! Let's dive into the fascinating world of trigonometry and explore the function y = 2sin(x - π/6) - 2. We're going to break down this function, figure out its key features, and then learn how to sketch its graph using some clever transformations. This is going to be fun, I promise! We'll cover finding the maximum and minimum values of the function. After that, we'll build the graph step by step, which is an important skill to learn. Understanding how to analyze and visualize trigonometric functions is super useful. It's not just about memorizing formulas; it's about seeing how these functions behave and how they transform. Let's get started!

a) Нахождение Наибольшего и Наименьшего Значений Функции

Okay, so first things first: let's find the maximum and minimum values of y = 2sin(x - π/6) - 2. The key here is to remember the basics of the sine function, sin(x). We know that the sine function oscillates between -1 and 1. This means the value of sin(x) is always within this range, so -1 ≤ sin(x) ≤ 1. The sine function is awesome because it repeats itself, always going up and down between these two limits. That's a super important thing to remember. Since we know the range of sin(x), we can use this information to determine the range of our specific function.

Now, let's consider the transformations in our function. We have 2sin(x - π/6) - 2. The '2' in front of the sine function is a vertical stretch, meaning it stretches the graph of the function sin(x) vertically by a factor of 2. So, instead of going between -1 and 1, the function 2sin(x) will oscillate between -2 and 2. The x - π/6 inside the sine function represents a horizontal shift (or phase shift). It shifts the graph π/6 units to the right. However, this horizontal shift doesn't affect the maximum or minimum values, it only changes where those values occur. Finally, the '-2' at the end is a vertical shift. This shifts the entire graph downward by 2 units. This is the crucial step for finding the maximum and minimum. Because the function is stretched by a factor of 2, and then shifted down by 2 units, we can calculate the maximum and minimum values.

To find the maximum value, we take the maximum value of the sine function (which is 1), multiply it by 2 (because of the vertical stretch), and then subtract 2 (because of the vertical shift). Thus, the maximum value is 2(1) - 2 = 0. For the minimum value, we take the minimum value of the sine function (-1), multiply it by 2, and subtract 2, resulting in 2(-1) - 2 = -4. Therefore, the maximum value of the function is 0, and the minimum value is -4. So, we've successfully found the greatest and least values of the function!

b) Построение Графика Функции с Помощью Преобразований

Alright, now for the fun part: sketching the graph! We're going to build this graph step-by-step, using transformations of the basic sine function, sin(x). Remember, understanding transformations is key to graphing trig functions. It's like having a set of instructions to morph the original function into something new. Let's break it down into easy-to-understand steps.

1. Start with the Basic Sine Function: Begin with the graph of y = sin(x). This is the foundation. It's a wave that oscillates between -1 and 1, crossing the x-axis at multiples of π (π, 2π, 3π, etc.). It starts at 0, goes up to 1, back down to -1, and then back to 0 over one complete cycle (2π). Knowing the key points and shape of this basic graph is super helpful.
2. Horizontal Shift: Now, let's consider the x - π/6 part. This is a horizontal shift (or phase shift). The graph will shift to the right by π/6 units. This means every point on the graph of y = sin(x) will move π/6 units to the right. The key points where the graph crosses the x-axis and reaches its maximum and minimum values will be affected by this shift. For example, the point where the sine wave usually starts (0, 0) will now become (π/6, 0). The entire wave pattern shifts in the positive x direction.
3. Vertical Stretch: Next, let's tackle the vertical stretch, the '2' in front of the sine function: 2sin(x - π/6). This means the graph of sin(x - π/6) will be stretched vertically by a factor of 2. The amplitude of the wave, which is the distance from the midline to the crest or trough, changes. The maximum value becomes 2, and the minimum value becomes -2. This makes the wave 'taller'.
4. Vertical Shift: Finally, we have the '-2' at the end: 2sin(x - π/6) - 2. This is a vertical shift. It shifts the entire graph downward by 2 units. This means every point on the graph will move down by 2 units. The x-intercepts, the maximum, and the minimum points all change positions. The midline of the wave, which was previously the x-axis, now becomes the line y = -2. The maximum value of the function is now 0, and the minimum value is -4.

By following these steps, you can accurately sketch the graph of y = 2sin(x - π/6) - 2. Remember to mark the key points, such as the x-intercepts, maximum points, and minimum points, to make your graph as clear as possible. Practice makes perfect, so try graphing a few more trigonometric functions, and you'll become a pro in no time! Keep in mind the order of operations, paying close attention to which transformations are applied first. This will help you get the correct final graph. It's a fun and rewarding process.