Plotting Y=cos(x) Graph With Table & Discussing Y=3sin(x)

Hey guys! Today, we're diving into the fascinating world of trigonometric functions, specifically plotting the graph of y = cos(x) using a table of values and then discussing the behavior of y = 3sin(x). Understanding these graphs is crucial in algebra and trigonometry, as they pop up in various applications, from physics to engineering. So, let’s break it down step by step and make sure you're confident with these concepts.

Plotting y = cos(x) using a Table

First off, let’s tackle y = cos(x). To plot this graph effectively, we’ll create a table of values. This table will give us specific points on the graph, which we can then connect to visualize the cosine function. The key here is to choose values of x that are easy to work with and cover a full cycle of the cosine function, which is 2π (or 360 degrees if you prefer working in degrees).

Creating the Table of Values

We'll use common angles like 0, π/6, π/4, π/3, π/2, and so on, up to 2π. These angles are strategic because their cosine values are well-known and easy to calculate. Remember, x represents the angle in radians, and y will be the cosine of that angle. Let’s get this table going!

Plotting the Points

Now that we have our table, it’s time to plot these points on a graph. The x-axis will represent the angle in radians, and the y-axis will represent the value of cos(x). Each row in our table gives us a coordinate (x, cos(x)) to plot. For example, (0, 1), (π/6, 0.87), and so on.

1. Set up your axes: Draw your x and y axes. Make sure you have enough space to represent the values from 0 to 2π on the x-axis and -1 to 1 on the y-axis. This is because the cosine function oscillates between -1 and 1.
2. Mark the key points on the x-axis: Mark the key angles like π/2, π, 3π/2, and 2π. These will help you space out your points evenly.
3. Plot each point: Carefully plot each point from the table onto your graph. Take your time to ensure accuracy.

Connecting the Dots

Once you’ve plotted all the points, the next step is to connect them. The cosine function is a smooth, continuous curve, so you’ll want to draw a smooth line through the points, rather than connecting them with straight lines. The graph should look like a wave oscillating between -1 and 1. This wave is the classic cosine curve!

Characteristics of y = cos(x) Graph

• Amplitude: The amplitude of the cosine function is 1. This means the maximum displacement from the x-axis is 1 (the highest point is at y = 1) and the minimum displacement is -1 (the lowest point is at y = -1).
• Period: The period of the cosine function is 2π. This means the function repeats itself every 2π radians. You can see one complete cycle of the wave between 0 and 2π on your graph.
• Symmetry: The cosine function is an even function, which means it's symmetric about the y-axis. Mathematically, this means cos(x) = cos(-x). If you fold the graph along the y-axis, the two halves will match up.
• Key Points: The graph starts at (0, 1), reaches its minimum at (π, -1), and returns to its maximum at (2π, 1). It crosses the x-axis at π/2 and 3π/2.

Discussing y = 3sin(x)

Now, let’s switch gears and discuss y = 3sin(x). This function is a variation of the standard sine function, y = sin(x), with a crucial difference: the coefficient 3 in front of the sine function. This coefficient affects the amplitude of the graph. Understanding this transformation is key to mastering trigonometric functions.

Understanding the Transformation

The function y = 3sin(x) is a vertical stretch of the sine function y = sin(x) by a factor of 3. What does this mean graphically? It means that the sine wave will be stretched vertically, making it taller.

Amplitude

In the standard sine function, y = sin(x), the amplitude is 1, meaning the graph oscillates between -1 and 1. However, in y = 3sin(x), the amplitude is 3. This means the graph will now oscillate between -3 and 3. The maximum value of the function is 3, and the minimum value is -3.

Period

The period of the function y = 3sin(x) remains the same as the standard sine function, which is 2π. The vertical stretch does not affect the period. This means the graph will still complete one full cycle within the interval [0, 2π].

Key Points and Graph Shape

To visualize the graph, let's consider some key points:

• When x = 0, y = 3sin(0) = 0. So the graph starts at (0, 0).
• When x = π/2, y = 3sin(π/2) = 3. This is the maximum point of the wave.
• When x = π, y = 3sin(π) = 0. The graph crosses the x-axis at π.
• When x = 3π/2, y = 3sin(3π/2) = -3. This is the minimum point of the wave.
• When x = 2π, y = 3sin(2π) = 0. The graph completes one full cycle and returns to 0.

Plotting these points and connecting them smoothly will give you a sine wave that oscillates between -3 and 3. It looks like the standard sine wave, but it’s been stretched vertically.

Comparing y = 3sin(x) to y = sin(x)

• Amplitude: The amplitude of y = 3sin(x) is 3, while the amplitude of y = sin(x) is 1. This is the main difference.
• Period: Both functions have the same period, 2π.
• Shape: Both functions have the same basic sine wave shape, but y = 3sin(x) is taller.
• Zeros: Both functions have zeros at the same points: x = 0, π, 2π, etc.

Applications and Implications

Understanding how coefficients affect trigonometric functions is crucial in many real-world applications. For instance:

• Physics: In physics, the sine and cosine functions are used to model oscillations and waves. The amplitude represents the maximum displacement of the wave, so a function like y = 3sin(x) could represent a wave with a greater intensity or displacement than y = sin(x).
• Engineering: Engineers use trigonometric functions in signal processing and control systems. The amplitude can represent the strength of a signal, and modifying the amplitude can be critical in designing systems that respond appropriately to different inputs.

Generalizing the Concept

More generally, for any function of the form y = A sin(x) or y = A cos(x), the amplitude is given by the absolute value of A. If A is greater than 1, the graph is stretched vertically; if A is between 0 and 1, the graph is compressed vertically. If A is negative, the graph is also reflected across the x-axis.

Conclusion

So, there you have it! We’ve plotted the graph of y = cos(x) using a table of values and discussed the implications of the transformation in y = 3sin(x). Remember, the key takeaways are:

• Creating a table of values is a great way to plot trigonometric functions accurately.
• The amplitude of a sine or cosine function affects its vertical stretch.
• Understanding these transformations helps in various real-world applications.

I hope this breakdown was helpful! Keep practicing plotting graphs and exploring different trigonometric functions. You’ll be a pro in no time. Keep rocking algebra, guys!