Problem 4 - Algebra 2 and Trigonometry

Question

What is the maximum value of \(y\) on the graph of \(y=\cos x ?\)

Step-by-Step Solution

Answer

The maximum value of \(y\) on the graph of \(y = \cos x\) is 1.

Step 1: Understanding the Graph of Cosine

The graph of the function \(y = \cos x\) is a periodic waveform that oscillates between -1 and 1. It is a continuous and smooth curve.

Step 2: Identify the Maximum Value

The maximum value of the cosine function, \(y = \cos x\), on its graph occurs whenever \(\cos x = 1\). This is because cosine values range from -1 to 1.

Step 3: Determine the Angle for Maximum Cosine

The cosine function \(\cos x = 1\) occurs at specific angles, namely \(x = 0, 2\pi, 4\pi, \ldots\) or any multiple of \(2\pi\). These are the points at which the cosine function reaches its peak.

Step 4: Final Conclusion

Since the maximum value of \(\cos x\) is 1 at certain angles, the maximum value of \(y\) on the graph of \(y = \cos x\) is 1.

Key Concepts

Periodic Function

A periodic function is a type of function that repeats its values in regular intervals or periods. In simpler terms, if you shift the graph of a periodic function horizontally by a certain distance, called the period, the graph looks exactly the same. One of the most common examples of a periodic function is the sine or cosine function.

For instance, the function \( y = \cos x \) has a period of \( 2\pi \), meaning that every \( 2\pi \) units along the x-axis, the entire pattern of the graph repeats itself. This feature is significant because it allows us to predict the behavior of the function over any interval.

Periodic functions have practical applications in various fields such as engineering, physics, and music, essentially anywhere waveforms or patterns repeat over time.

Cosine Function

The cosine function is a fundamental trigonometric function represented as \( \cos x \). In terms of a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. However, when we graph \( y = \cos x \), it takes the shape of a wave which oscillates smoothly and is periodic with a period of \( 2\pi \).

At the starting point \( x = 0 \), \( \cos x \) equals 1, which is its maximum value. As you move along the x-axis from \( 0 \) to \( 2\pi \), the cosine value decreases, reaching -1 at \( x = \pi \), then it increases again back to 1.

• The cosine curve goes through a complete cycle every \( 2\pi \), hitting its peak points and lowest points consistently.
• Cosine is an even function, meaning that \( \cos(-x) = \cos(x) \), this symmetry is evident in its graph.

This predictable behavior is useful in applications involving cycles and oscillations.

Range of Cosine Function

The range of a function refers to the set of all possible output values (y-values) it can produce. For the cosine function, \( y = \cos x \), the range is remarkably simple yet crucial to understanding its behavior.

The cosine function yields values only between -1 and 1, inclusive. This means that regardless of the x-coordinate you choose, the y-coordinate when you input into \( \cos x \) will always fall within this interval.

• The maximum value it reaches is 1, occurring at angles like \( x = 0, 2\pi, 4\pi, \ldots \).
• The minimum value it reaches is -1, occurring at angles like \( x = \pi, 3\pi, 5\pi, \ldots \).

These consistent maximum and minimum y-values mean the graph of the cosine function will always be bounded within a horizontal strip from -1 to 1 on the y-axis, creating its characteristic wave-like appearance. This bounded nature makes the cosine function a staple in modeling wave patterns.