Max/Min & Graph Of Y = 2sin(x-7)-2: Explained!

Alright, guys, let's break down this function: y = 2sin(x - 7) - 2. We're going to figure out its maximum and minimum values and then sketch its graph using transformations. Buckle up; it's gonna be a fun ride!

Finding the Maximum and Minimum Values

First, let's tackle finding the maximum and minimum values of the function. The key here is understanding the sine function. We know that the sine function, sin(x), oscillates between -1 and 1. That is, for any value of x, -1 ≤ sin(x) ≤ 1. Understanding this behavior is crucial to determining the range of our given function.

Understanding the Sine Function's Range: The standard sine function, sin(x), always produces values between -1 and 1. This is because it represents the y-coordinate of a point on the unit circle as the angle x changes. The maximum y-coordinate is 1 (at 90 degrees or π/2 radians), and the minimum y-coordinate is -1 (at 270 degrees or 3π/2 radians). This inherent property of the sine function is the foundation for finding the maximum and minimum values of more complex sinusoidal functions.

Now, let's consider the transformations applied to the sine function in our equation: y = 2sin(x - 7) - 2. The 2 in front of the sine function stretches the graph vertically. Instead of oscillating between -1 and 1, 2sin(x) will oscillate between -2 and 2. Think of it as amplifying the sine wave. The (x - 7) inside the sine function shifts the graph horizontally by 7 units to the right. This shift doesn't affect the maximum or minimum values but changes the position of the graph along the x-axis. Finally, the - 2 at the end shifts the entire graph vertically downward by 2 units. This shift directly impacts the maximum and minimum values.

To find the maximum value, we start with the maximum value of sin(x), which is 1. Then we multiply it by 2 (due to the vertical stretch) to get 2. Finally, we subtract 2 (due to the vertical shift) to get the maximum value of our function: 2 * 1 - 2 = 0. So, the maximum value of the function y = 2sin(x - 7) - 2 is 0. This occurs when sin(x - 7) is at its maximum value of 1.

Similarly, to find the minimum value, we start with the minimum value of sin(x), which is -1. We multiply it by 2 to get -2 and then subtract 2 to get the minimum value of our function: 2 * (-1) - 2 = -4. Thus, the minimum value of the function y = 2sin(x - 7) - 2 is -4. This occurs when sin(x - 7) is at its minimum value of -1.

In summary, the maximum value of the function y = 2sin(x - 7) - 2 is 0, and the minimum value is -4. These values define the range of the function, providing a clear understanding of its vertical boundaries. Remember, the transformations applied to the sine function—vertical stretch and vertical shift—directly influence these maximum and minimum values.

Graphing the Function Using Transformations

Okay, let's sketch the graph of y = 2sin(x - 7) - 2 using transformations. We'll start with the basic sine function and apply each transformation step-by-step.

1. Start with the Basic Sine Function: y = sin(x)

The graph of y = sin(x) is a wave that oscillates between -1 and 1. It passes through the origin (0, 0), and its period is 2π. That means it completes one full cycle every 2π units along the x-axis. Key points to remember are (0, 0), (π/2, 1), (π, 0), (3π/2, -1), and (2π, 0). This is our starting point.

2. Horizontal Shift: y = sin(x - 7)

Replacing x with (x - 7) shifts the graph horizontally. Specifically, y = sin(x - 7) shifts the graph of y = sin(x) to the right by 7 units. This is a horizontal translation. Every point on the original sine wave moves 7 units to the right. For example, the point (0, 0) on y = sin(x) moves to (7, 0) on y = sin(x - 7). This shift doesn't change the shape or amplitude of the wave, only its position along the x-axis.

3. Vertical Stretch: y = 2sin(x - 7)

Multiplying the sine function by 2 stretches the graph vertically. The graph of y = 2sin(x - 7) is the graph of y = sin(x - 7) stretched vertically by a factor of 2. This means that the amplitude of the wave doubles. Instead of oscillating between -1 and 1, it now oscillates between -2 and 2. The key points are now (7, 0), (7 + π/2, 2), (7 + π, 0), (7 + 3π/2, -2), and (7 + 2π, 0). The peaks and troughs of the wave are twice as high and twice as low, respectively.

4. Vertical Shift: y = 2sin(x - 7) - 2

Finally, subtracting 2 from the entire function shifts the graph vertically downward by 2 units. The graph of y = 2sin(x - 7) - 2 is the graph of y = 2sin(x - 7) shifted down by 2 units. Every point on the wave moves 2 units downward. The key points become (7, -2), (7 + π/2, 0), (7 + π, -2), (7 + 3π/2, -4), and (7 + 2π, -2). This vertical shift changes the midline of the wave from y = 0 to y = -2.

By applying these transformations step-by-step, we can accurately sketch the graph of y = 2sin(x - 7) - 2. Start with the basic sine wave, shift it horizontally, stretch it vertically, and then shift it vertically. Each transformation alters the graph in a predictable way, making it easier to visualize and understand the final function.

Putting It All Together

So, to recap:

• Maximum Value: 0
• Minimum Value: -4
• Graph: Start with y = sin(x), shift right by 7, stretch vertically by 2, and shift down by 2.

And there you have it! You've successfully found the maximum and minimum values of the function and learned how to graph it using transformations. Keep practicing, and you'll become a pro at these types of problems!

Understanding these concepts is super helpful, and I hope this explanation made it crystal clear for you all! Keep rocking those math problems! 🚀