https://file.aiquickdraw.com/mathbot/file/1729832241376cw42nve5.jpg

Let's work through the problem step by step.

### Part (a) — Finding the amplitude, period, and horizontal shift:
1. **Amplitude**: The amplitude is the distance from the midline of the cosine curve to its peak (or trough). From the graph, the peak occurs at \( y = \frac{1}{10} \) and the trough at \( y = -\frac{1}{10} \). Therefore, the amplitude is:
   \[
   \text{Amplitude} = \frac{1}{10}
   \]

2. **Period**: The period of a cosine function is the horizontal distance between two consecutive peaks (or troughs). The graph shows that a full cycle starts at \( x = -\frac{\pi}{4} \) and ends at \( x = \frac{\pi}{4} \), so the period is:
   \[
   \text{Period} = \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{2}
   \]

3. **Horizontal Shift**: The standard cosine function starts at its peak when \( x = 0 \). In this graph, the peak occurs at \( x = 0 \), so there is no horizontal shift:
   \[
   \text{Horizontal shift} = 0
   \]

### Part (b) — Equation of the cosine curve:
The general form of the cosine function is:
\[
y = a \cos(k(x - b))
\]
Where:
- \( a \) is the amplitude,
- \( k \) is related to the period by \( k = \frac{2\pi}{\text{Period}} \),
- \( b \) is the horizontal shift.

From part (a):
- Amplitude \( a = \frac{1}{10} \),
- Period \( \frac{\pi}{2} \), so \( k = \frac{2\pi}{\frac{\pi}{2}} = 4 \),
- Horizontal shift \( b = 0 \).

Thus, the equation is:
\[
y = \frac{1}{10} \cos(4x)
\]

Let me know if you would like further details or have any questions. Here are some related questions:

1. What is the significance of the negative amplitude in a cosine graph?
2. How would the equation change if there was a vertical shift?
3. How does the period of a cosine graph affect its frequency?
4. Can you explain how phase shifts work in cosine functions?
5. How would the graph change if the period was doubled?

**Tip**: Remember that the factor \( k \) inside the cosine function is directly related to the period through \( k = \frac{2\pi}{\text{Period}} \). This helps you adjust the horizontal stretch of the graph!

Find the amplitude, period, horizontal shift, and write the equation for the given cosine curve.

Learn how to find the amplitude, period, and horizontal shift of a cosine curve, and write the corresponding equation. This problem involves key trigonometry concepts and is 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