Graph The Function Over The Interval \[$[-2 \pi, 2 \pi]\$\].Function: \[$y = 4 \cos X\$\]Give The Amplitude. The Amplitude Is \[$\square\$\].(Simplify Your Answer. Type An Integer Or A Fraction.)

Introduction

Graphing trigonometric functions is a crucial aspect of mathematics, particularly in calculus and analysis. In this article, we will focus on graphing the function $y = 4 \cos x$ over the interval $[-2 \pi, 2 \pi]$. We will also discuss the concept of amplitude and how it relates to the given function.

Understanding the Function

The given function is $y = 4 \cos x$. This is a cosine function with a coefficient of 4, which represents the amplitude of the function. The amplitude of a cosine function is the maximum value that the function attains. In this case, the amplitude is 4.

Graphing the Function

To graph the function $y = 4 \cos x$, we need to understand the behavior of the cosine function. The cosine function is a periodic function that oscillates between -1 and 1. When we multiply the cosine function by 4, the amplitude of the function increases to 4.

The graph of the function $y = 4 \cos x$ over the interval $[-2 \pi, 2 \pi]$ is a sinusoidal curve that oscillates between -4 and 4. The graph has a period of $2 \pi$, which means that it repeats itself every $2 \pi$ units.

Key Features of the Graph

The graph of the function $y = 4 \cos x$ has several key features that are worth noting:

• Amplitude: The amplitude of the function is 4, which is the maximum value that the function attains.
• Period: The period of the function is $2 \pi$, which means that it repeats itself every $2 \pi$ units.
• X-intercepts: The x-intercepts of the function are at $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$, $x = \frac{3\pi}{2}$, and $x = -\frac{3\pi}{2}$.
• Y-intercept: The y-intercept of the function is at $y = 4$.

Graphing the Function using Technology

Graphing the function $y = 4 \cos x$ using technology is a straightforward process. We can use a graphing calculator or a computer algebra system (CAS) to graph the function.

Step 1: Enter the Function

Enter the function $y = 4 \cos x$ into the graphing calculator or CAS.

Step 2: Set the Window

Set the window to the desired interval, in this case, $[-2 \pi, 2 \pi]$.

Step 3: Graph the Function

Graph the function using the graphing calculator or CAS.

Step 4: Analyze the Graph

Analyze the graph to identify the key features of the function, such as the amplitude, period, x-intercepts, and y-intercept.

Conclusion

Graphing the function $y = 4 \cos x$ over the interval $[-2 \pi, 2 \pi]$ is a straightforward process that can be accomplished using technology. The graph of the function has several key features, including an amplitude of 4, a period of $2 \pi$, x-intercepts at $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$, $x = \frac{3\pi}{2}$, and $x = -\frac{3\pi}{2}$, and a y-intercept at $y = 4$.

Amplitude

The amplitude of a cosine function is the maximum value that the function attains. In the case of the function $y = 4 \cos x$, the amplitude is 4.

Final Answer

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Q&A: Graphing Trigonometric Functions

Q: What is the amplitude of the function $y = 4 \cos x$?

A: The amplitude of the function $y = 4 \cos x$ is 4. This is the maximum value that the function attains.

Q: What is the period of the function $y = 4 \cos x$?

A: The period of the function $y = 4 \cos x$ is $2 \pi$. This means that the function repeats itself every $2 \pi$ units.

Q: What are the x-intercepts of the function $y = 4 \cos x$?

A: The x-intercepts of the function $y = 4 \cos x$ are at $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$, $x = \frac{3\pi}{2}$, and $x = -\frac{3\pi}{2}$.

Q: What is the y-intercept of the function $y = 4 \cos x$?

A: The y-intercept of the function $y = 4 \cos x$ is at $y = 4$.

Q: How do I graph the function $y = 4 \cos x$ using technology?

A: To graph the function $y = 4 \cos x$ using technology, follow these steps:

1. Enter the function $y = 4 \cos x$ into the graphing calculator or CAS.
2. Set the window to the desired interval, in this case, $[-2 \pi, 2 \pi]$.
3. Graph the function using the graphing calculator or CAS.
4. Analyze the graph to identify the key features of the function.

Q: What are some common mistakes to avoid when graphing trigonometric functions?

A: Some common mistakes to avoid when graphing trigonometric functions include:

• Not setting the window to the correct interval.
• Not using the correct function or equation.
• Not analyzing the graph to identify the key features of the function.
• Not using technology to graph the function, such as a graphing calculator or CAS.

Q: How do I determine the amplitude of a trigonometric function?

A: To determine the amplitude of a trigonometric function, look for the coefficient of the sine or cosine term. The amplitude is the absolute value of this coefficient.

Q: How do I determine the period of a trigonometric function?

A: To determine the period of a trigonometric function, look for the coefficient of the sine or cosine term. The period is the reciprocal of the absolute value of this coefficient.

Conclusion

Graphing trigonometric functions is a crucial aspect of mathematics, particularly in calculus and analysis. By understanding the key features of trigonometric functions, such as amplitude and period, we can graph these functions with ease. Remember to use technology, such as graphing calculators or CAS, to graph trigonometric functions and analyze their key features.

Common Trigonometric Functions

Here are some common trigonometric functions and their key features:

• Sine Function: $y = \sin x$
  • Amplitude: 1
  • Period: $2 \pi$
  • X-intercepts: $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$, $x = \frac{3\pi}{2}$, and $x = -\frac{3\pi}{2}$
  • Y-intercept: $y = 0$
• Cosine Function: $y = \cos x$
  • Amplitude: 1
  • Period: $2 \pi$
  • X-intercepts: $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$, $x = \frac{3\pi}{2}$, and $x = -\frac{3\pi}{2}$
  • Y-intercept: $y = 1$
• Tangent Function: $y = \tan x$
  • Amplitude: undefined
  • Period: $\pi$
  • X-intercepts: $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$, $x = \frac{3\pi}{2}$, and $x = -\frac{3\pi}{2}$
  • Y-intercept: $y = 0$

Final Answer

The final answer is: $\boxed{4}$