How do you graph y=1-cos x ?

To graph the equation \( y = 1 - \cos x \), we first need to understand the behavior of the cosine function and how it transforms when we apply the equation. The cosine function oscillates between -1 and 1, which will significantly influence the shape of our graph.

Understanding the Cosine Function

The basic cosine function, \( \cos x \), has a period of \( 2\pi \). This means it repeats its values every \( 2\pi \) units. The key points of \( \cos x \) are:

• At \( x = 0 \), \( \cos(0) = 1 \)
• At \( x = \frac{\pi}{2} \), \( \cos\left(\frac{\pi}{2}\right) = 0 \)
• At \( x = \pi \), \( \cos(\pi) = -1 \)
• At \( x = \frac{3\pi}{2} \), \( \cos\left(\frac{3\pi}{2}\right) = 0 \)
• At \( x = 2\pi \), \( \cos(2\pi) = 1 \)

Transforming the Function

Now, when we apply the transformation \( y = 1 - \cos x \), we need to analyze how this affects the output values:

• At \( x = 0 \): \( y = 1 - \cos(0) = 1 - 1 = 0 \)
• At \( x = \frac{\pi}{2} \): \( y = 1 - \cos\left(\frac{\pi}{2}\right) = 1 - 0 = 1 \)
• At \( x = \pi \): \( y = 1 - \cos(\pi) = 1 - (-1) = 2 \)
• At \( x = \frac{3\pi}{2} \): \( y = 1 - \cos\left(\frac{3\pi}{2}\right) = 1 - 0 = 1 \)
• At \( x = 2\pi \): \( y = 1 - \cos(2\pi) = 1 - 1 = 0 \)

Sketching the Graph

With these points calculated, we can start sketching the graph:

• Plot the points: (0, 0), \( \left(\frac{\pi}{2}, 1\right) \), (π, 2), \( \left(\frac{3\pi}{2}, 1\right) \), and (2π, 0).
• Notice that the graph will have a minimum value of 0 at \( x = 0 \) and \( x = 2\pi \), and a maximum value of 2 at \( x = \pi \).
• The graph will be symmetrical about the line \( x = \pi \), creating a wave-like shape that peaks at \( y = 2 \) and dips back down to \( y = 0 \).

Characteristics of the Graph

Some additional features to note:

• The graph is continuous and smooth, reflecting the continuous nature of the cosine function.
• It has a period of \( 2\pi \), just like the cosine function, meaning it will repeat its pattern every \( 2\pi \) units.
• The range of the function is from 0 to 2, as the minimum value occurs at the peaks of the cosine function and the maximum occurs at its troughs.

By following these steps, you can effectively graph \( y = 1 - \cos x \) and understand its behavior in relation to the cosine function. If you have any further questions or need clarification on specific points, feel free to ask!