Sine Function Amplitude, Period, And Shifts Explained

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of sine functions and tackling a real-world problem that will put your knowledge to the test. We'll break down the anatomy of a sine function, explore its key components, and then apply our understanding to find the y-value for a specific x-value. So, buckle up and get ready for a mathematical adventure!

Understanding Sine Functions: The Building Blocks

At its core, a sine function is a trigonometric function that describes a smooth, periodic oscillation. Imagine a wave gracefully undulating through space – that's essentially what a sine function represents. The general form of a sine function is given by:

y = A * sin(B(x - C)) + D

Where:

• A represents the amplitude, which determines the vertical stretch of the function.
• B is related to the period, which is the length of one complete cycle of the wave. The period is calculated as $2\pi/B$.
• C represents the horizontal shift (also known as the phase shift), which indicates how much the function is shifted left or right.
• D represents the vertical shift, which determines how much the function is shifted up or down.

Let's delve deeper into each of these components to gain a more intuitive understanding.

Amplitude: The Height of the Wave

The amplitude, denoted by A, is the distance from the midline of the function to its maximum or minimum point. It essentially dictates how "tall" the wave is. A larger amplitude means a taller wave, while a smaller amplitude means a shorter wave. Think of it like the volume knob on a stereo – the higher the amplitude, the louder the sound (or, in this case, the taller the wave).

To visualize this, imagine a standard sine wave, which oscillates between -1 and 1. If we double the amplitude, the wave will now oscillate between -2 and 2, effectively stretching it vertically. Conversely, if we halve the amplitude, the wave will oscillate between -0.5 and 0.5, compressing it vertically.

In our problem, we're given that the sine function has an amplitude of 3. This means that the wave will oscillate between -3 and 3 relative to its midline. This is a crucial piece of information that we'll use later to determine the y-value of the function.

Period: The Length of a Cycle

The period, as we mentioned earlier, is the length of one complete cycle of the wave. It's the distance along the x-axis it takes for the function to repeat its pattern. The period is inversely proportional to the value of B in the general equation. A larger value of B means a shorter period, and a smaller value of B means a longer period.

The standard sine function, y = sin(x), has a period of $2\pi$. This means that the wave completes one full cycle from 0 to $2\pi$. If we want to change the period, we need to adjust the value of B. For example, if we want to double the period, we need to halve the value of B, and vice versa.

In our problem, we're given that the sine function has a period of 6π. This is significantly longer than the standard period of $2\pi$. To achieve this elongated period, the value of B must be smaller than 1. We can calculate B using the formula: B = $2\pi/Period$, so B = $2\pi/6\pi = 1/3$.

Horizontal Shift: Sliding the Wave Left or Right

The horizontal shift, denoted by C, determines how much the function is shifted left or right along the x-axis. It's also known as the phase shift. A positive value of C shifts the function to the right, while a negative value of C shifts the function to the left. It's important to note that the shift is the opposite of what you might intuitively expect.

For example, if C is $π/2$, the function is shifted $π/2$ units to the right. This means that the starting point of the wave is no longer at x = 0, but at x = $π/2$. The horizontal shift is crucial for aligning the sine function with specific data or conditions.

In our problem, we're given that the sine function has a horizontal shift of $3\pi/2$. This means that the wave is shifted $3\pi/2$ units to the right. This shift will affect the x-coordinate at which the function reaches its maximum, minimum, and zero values.

Vertical Shift: Moving the Wave Up or Down

The vertical shift, denoted by D, determines how much the function is shifted up or down along the y-axis. It simply adds a constant value to the entire function, effectively raising or lowering the wave. A positive value of D shifts the function upwards, while a negative value of D shifts the function downwards.

For example, if D is 2, the entire sine wave is shifted 2 units upwards. This means that the midline of the wave is now at y = 2 instead of y = 0. The vertical shift is useful for adjusting the vertical position of the sine function to match a specific context.

In our problem, we're given that the sine function has a vertical shift of -1. This means that the entire wave is shifted 1 unit downwards. The midline of the wave will now be at y = -1, and the function will oscillate between -4 and 2.

Putting It All Together: The Equation of Our Sine Function

Now that we've dissected the components of a sine function, let's construct the equation for the specific function described in our problem. We know that:

• Amplitude (A) = 3
• Period = 6π, so B = 1/3
• Horizontal shift (C) = $3\pi/2$
• Vertical shift (D) = -1

Plugging these values into the general form of a sine function, we get:

y = 3 * sin((1/3)(x - 3π/2)) - 1

This is the equation that represents the sine function with the given characteristics. Now, we're ready to tackle the final part of the problem: finding the y-value when x = 2π.

Finding the y-value: A Step-by-Step Solution

To find the y-value when x = 2π, we simply substitute 2π for x in our equation:

y = 3 * sin((1/3)(2π - 3π/2)) - 1

Let's simplify this step by step:

1. Simplify the expression inside the parentheses:

   2π - 3π/2 = 4π/2 - 3π/2 = π/2
   

2. Multiply by 1/3:

   (1/3)(π/2) = π/6
   

3. Evaluate the sine function:

   sin(π/6) = 1/2
   

4. Multiply by 3:

   3 * (1/2) = 3/2
   

5. Subtract 1:

   3/2 - 1 = 1/2
   

Therefore, the y-value of the function when x = 2π is 1/2.

Conclusion: Mastering Sine Functions

We've successfully navigated the world of sine functions, breaking down their components and applying our knowledge to solve a real-world problem. Guys, remember, understanding the amplitude, period, horizontal shift, and vertical shift is key to mastering these fascinating functions. By carefully analyzing each component, we can accurately predict the behavior of sine waves and use them to model a wide range of phenomena in the world around us. So, keep practicing, keep exploring, and keep unlocking the power of mathematics!

Therefore, the final answer is 1/2