How do you determine the period of the graph of $$ y = A\cos(Bx) $$ where A and B are constants?

Question

UpStudy AI · Accepted Answer

To determine the **period** of the graph of the function $$ y = A\cos(Bx) $$, where $$ A $$ and $$ B $$ are constants, you can follow these steps:

### Understanding the Components

1. **Amplitude ($$ A $$)**:
   - $$ A $$ affects the **vertical stretch** or **compression** of the graph.
   - It does **not** influence the period.

2. **Angular Frequency ($$ B $$)**:
   - $$ B $$ affects the **horizontal stretch** or **compression** of the graph.
   - It directly influences the **period** of the cosine function.

### Determining the Period

The **standard period** of the cosine function $$ y = \cos(x) $$ is $$ 2\pi $$. When the function is modified to $$ y = \cos(Bx) $$, the period changes based on the value of $$ B $$.

The formula to find the period ($$ T $$) of $$ y = A\cos(Bx) $$ is:

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

- **Explanation**:
  - **$$ 2\pi $$** is the period of the basic cosine function.
  - **$$ B $$** modifies how quickly the cosine function completes one full cycle.
  - **$$ |B| $$** ensures the period is a positive value, as frequency is inherently positive.

### Example

Suppose you have the function $$ y = 3\cos(4x) $$:

1. **Identify $$ B $$**:
   - Here, $$ B = 4 $$.

2. **Apply the Period Formula**:
   $$
   T = \frac{2\pi}{|4|} = \frac{2\pi}{4} = \frac{\pi}{2}
   $$

3. **Interpretation**:
   - The period of $$ y = 3\cos(4x) $$ is $$ \frac{\pi}{2} $$.
   - This means the function completes one full cycle every $$ \frac{\pi}{2} $$ units along the $$ x $$-axis.

### Graphical Interpretation

- **Larger $$ B $$**: Results in a **smaller period**, causing the graph to oscillate more rapidly.
- **Smaller $$ B $$**: Results in a **larger period**, causing the graph to oscillate more slowly.

### Summary

To find the period of $$ y = A\cos(Bx) $$:

1. **Identify the value of $$ B $$** in the function.
2. **Use the period formula**: $$ T = \frac{2\pi}{|B|} $$.
3. **Calculate** to obtain the period.

This formula ensures you can determine how the graph of the cosine function behaves horizontally based on the value of $$ B $$.
The period of $$ y = A\cos(Bx) $$ is $$ \frac{2\pi}{|B|} $$.

How do you determine the period of the graph of where A and B are constants?

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Solución

Understanding the Components

Determining the Period

Example

Graphical Interpretation

Summary

The Deep Dive

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