https://file.aiquickdraw.com/mathbot/file/1725645440199bi735xvl.jpg

From the image, I see that the problem is asking to find the formula for the graph of a trigonometric function, likely a sine or cosine function based on the shape of the graph (a sinusoidal wave). The image shows that the graph has two full peaks and troughs, indicating periodic behavior typical of sine or cosine functions.

### Key characteristics of the graph:
1. **Amplitude**: The graph has peaks and troughs that reach approximately the same height above and below the x-axis. So, the amplitude is roughly 2 units.
2. **Period**: The distance over which the wave repeats itself (i.e., the period) seems to be approximately 6π, as the graph completes two cycles between 0 and 6π on the x-axis.
3. **Midline**: The graph appears centered on the x-axis, so the vertical shift (midline) is zero.
4. **Phase shift**: The graph starts at a maximum at x = 0, suggesting it might represent a cosine function rather than sine (cosine starts at the maximum when there is no phase shift).

### General form of a cosine function:
The general form of a cosine function is:
\[ f(x) = A \cos(Bx - C) + D \]
Where:
- \( A \) is the amplitude.
- \( B \) determines the period (\( \text{Period} = \frac{2\pi}{B} \)).
- \( C \) is the phase shift.
- \( D \) is the vertical shift.

### Based on the graph:
- **Amplitude (A)**: 2.
- **Period**: 6π, so \( B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{6\pi} = \frac{1}{3} \).
- **Phase shift (C)**: There appears to be no horizontal shift, so \( C = 0 \).
- **Vertical shift (D)**: 0, since the graph is centered on the x-axis.

### The equation:
Thus, the function is likely:
\[
f(x) = 2 \cos\left(\frac{1}{3}x\right)
\]

Would you like more details or have any questions?

---

Here are five related questions you might find helpful:
1. How do you calculate the period of a trigonometric function?
2. What are the differences between sine and cosine functions?
3. How do amplitude and vertical shifts affect a graph?
4. What are phase shifts and how do they change the function’s behavior?
5. Can sine and cosine functions be combined to model more complex waves?

**Tip**: Remember that for cosine graphs, if the graph starts at a peak (maximum), it is often easier to model it with a cosine function rather than a sine function.

Find a formula for the graph of the trigonometric function shown in the given graph.

Learn how to find the formula for a trigonometric function based on its graph. This example covers key concepts such as amplitude, period, phase shift, and vertical shift, using a cosine function as a model. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation