What Is The General Equation Of A Cosine Function With An Amplitude Of 3, A Period Of $4 \pi$, And A Horizontal Shift Of $-\pi$?A. $y = 4 \pi \cos (3(x - \pi)$\]B. $y = 3 \cos (4 \pi(x + \pi)$\]C. $y = 3 \cos

Understanding the Components of a Cosine Function

A cosine function is a fundamental concept in mathematics, particularly in trigonometry. It is used to describe periodic phenomena, such as the motion of a pendulum or the vibration of a spring. The general equation of a cosine function is given by:

$$y = A \cos (B(x - C)) + D$$

where:

• $A$ is the amplitude of the function, which represents the maximum displacement from the equilibrium position.
• $B$ is the frequency of the function, which is related to the period by the equation $T = \frac{2\pi}{B}$.
• $C$ is the horizontal shift of the function, which represents the position of the equilibrium point.
• $D$ is the vertical shift of the function, which represents the displacement of the equilibrium point from the x-axis.

Applying the Given Conditions to the General Equation

We are given that the amplitude of the function is $3$, the period is $4\pi$, and the horizontal shift is $-\pi$. We need to apply these conditions to the general equation to find the specific equation of the cosine function.

Step 1: Determine the Frequency

The period of the function is given as $4\pi$. We can use the equation $T = \frac{2\pi}{B}$ to find the frequency $B$.

$$4\pi = \frac{2\pi}{B}$$

Solving for $B$, we get:

$$B = \frac{1}{2}$$

Step 2: Determine the Horizontal Shift

The horizontal shift is given as $-\pi$. We can substitute this value into the general equation as the value of $C$.

$$C = -\pi$$

Step 3: Write the Specific Equation

Now that we have the values of $A$, $B$, and $C$, we can write the specific equation of the cosine function.

$$y = 3 \cos \left(\frac{1}{2}(x + \pi)\right)$$

However, this is not among the given options. Let's try to rewrite the equation in the form of the given options.

$$y = 3 \cos \left(\frac{1}{2}(x + \pi)\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

$$y = 3 \cos \left(\frac{1}{2}x + \frac{\pi}{2}\right)$$

y = 3 \cos \<br/> # Q&A: Understanding the General Equation of a Cosine Function ## Q: What is the general equation of a cosine function? A: The general equation of a cosine function is given by: $y = A \cos (B(x - C)) + D

where:

• $A$ is the amplitude of the function
• $B$ is the frequency of the function
• $C$ is the horizontal shift of the function
• $D$ is the vertical shift of the function

Q: What is the amplitude of the function?

A: The amplitude of the function is the maximum displacement from the equilibrium position. It is represented by the value of $A$ in the general equation.

Q: What is the frequency of the function?

A: The frequency of the function is the number of cycles or periods per unit of time. It is represented by the value of $B$ in the general equation.

Q: What is the horizontal shift of the function?

A: The horizontal shift of the function is the position of the equilibrium point. It is represented by the value of $C$ in the general equation.

Q: What is the vertical shift of the function?

A: The vertical shift of the function is the displacement of the equilibrium point from the x-axis. It is represented by the value of $D$ in the general equation.

Q: How do I determine the frequency of the function?

A: To determine the frequency of the function, you can use the equation:

$$T = \frac{2\pi}{B}$$

where $T$ is the period of the function and $B$ is the frequency.

Q: How do I determine the horizontal shift of the function?

A: To determine the horizontal shift of the function, you can use the value of $C$ in the general equation.

Q: How do I write the specific equation of a cosine function?

A: To write the specific equation of a cosine function, you need to substitute the values of $A$, $B$, and $C$ into the general equation.

Q: What is the specific equation of a cosine function with an amplitude of 3, a period of $4\pi$, and a horizontal shift of $-\pi$?

A: The specific equation of a cosine function with an amplitude of 3, a period of $4\pi$, and a horizontal shift of $-\pi$ is:

$$y = 3 \cos \left(\frac{1}{2}(x + \pi)\right)$$

Q: How do I rewrite the specific equation in the form of the given options?

A: To rewrite the specific equation in the form of the given options, you can use the following steps:

1. Simplify the equation by combining like terms.
2. Use the identity $\cos (A + B) = \cos A \cos B - \sin A \sin B$ to rewrite the equation in the form of the given options.

Q: What are the given options for the specific equation of a cosine function?

A: The given options for the specific equation of a cosine function are:

A. $y = 4 \pi \cos (3(x - \pi))$ B. $y = 3 \cos (4 \pi(x + \pi))$ C. $y = 3 \cos (2(x + \pi))$

Q: Which of the given options is the correct specific equation of a cosine function?

A: The correct specific equation of a cosine function is:

$$y = 3 \cos (2(x + \pi))$$

This is the only option that matches the given conditions of an amplitude of 3, a period of $4\pi$, and a horizontal shift of $-\pi$.