https://file.aiquickdraw.com/mathbot/file/17286858634665l13elbf.jpg

To write the equation of a sine or cosine function for the given graph, let's analyze the key properties of the function:

### 1. Amplitude
The amplitude is the distance from the midline (y = 0 in this case) to the maximum or minimum point. Here, the maximum value is \( y = 4 \) and the minimum value is \( y = -4 \). Therefore, the amplitude is:
\[
A = 4
\]

### 2. Period
The period is the distance required for the function to complete one full cycle. Observing the graph, one full cycle starts at \( x = -\frac{\pi}{2} \), reaches the next peak at \( x = \frac{\pi}{2} \), and completes at \( x = \frac{3\pi}{2} \). The length of this cycle is:
\[
T = \frac{3\pi}{2} + \frac{\pi}{2} = 2\pi
\]
So, the period is \( 2\pi \).

### 3. Phase Shift
Since the graph passes through the origin \( (0, 0) \) and follows the form of a sine wave, there’s no phase shift. For a sine wave, this implies it starts at zero when \( x = 0 \), which aligns with the sine function.

### 4. Vertical Shift
There’s no vertical shift, as the midline of the function is \( y = 0 \).

### Equation Form
For a sine function with these characteristics, the general form is:
\[
y = A \sin(Bx)
\]
Where:
- \( A = 4 \), as derived above.
- \( B \) is determined by the period. Since the period \( T = 2\pi \), and the relationship between \( B \) and \( T \) is \( T = \frac{2\pi}{B} \), we get \( B = 1 \).

### Final Equation
Thus, the equation of the function is:
\[
y = 4 \sin(x)
\]

Alternatively, this graph could be described by a cosine function with a phase shift of \( \frac{\pi}{2} \):
\[
y = 4 \cos(x - \frac{\pi}{2})
\]

Either equation describes the graph accurately. Would you like any further details or have any questions?

Here are five related questions:
1. How would the equation change if the amplitude were halved?
2. What would the equation be if the function were vertically shifted up by 3 units?
3. How does a phase shift affect the appearance of the sine or cosine graph?
4. Can this graph be described with a tangent function? Why or why not?
5. What would be the effect on the graph if the period were \( \pi \) instead of \( 2\pi \)?

**Tip:** Remember, adjusting the amplitude or period directly affects the height and length of the cycles in sine and cosine graphs.

Write the equation of a sine or cosine function to describe the graph.

Learn how to derive the sine and cosine equations for a graph with an amplitude of 4 and a period of 2π. Understand key concepts like amplitude, period, and phase shift, and explore how the sine function can describe this graph.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation