Find the amplitude, period, and horizontal shift of the function.

*y* = 2 sin

5

6

*x* −

5𝜋

24

amplitude

period

horizontal shif

The given function is \( y = 2 \sin \left( \frac{5}{6}x - \frac{5\pi}{24} \right) \).

Let's break it down to find the **amplitude**, **period**, and **horizontal shift**.

### 1. **Amplitude**:
The amplitude of a sine function \( y = A \sin(Bx + C) \) is the absolute value of \( A \).

Here, \( A = 2 \), so the amplitude is:
\[
\text{Amplitude} = |2| = 2
\]

### 2. **Period**:
The period of a sine function is given by the formula:
\[
\text{Period} = \frac{2\pi}{|B|}
\]
where \( B \) is the coefficient of \( x \).

Here, \( B = \frac{5}{6} \), so the period is:
\[
\text{Period} = \frac{2\pi}{\frac{5}{6}} = 2\pi \times \frac{6}{5} = \frac{12\pi}{5}
\]

### 3. **Horizontal Shift**:
The horizontal shift (or phase shift) is given by the formula:
\[
\text{Horizontal shift} = -\frac{C}{B}
\]
where \( C \) is the constant inside the sine function.

Here, \( C = -\frac{5\pi}{24} \) and \( B = \frac{5}{6} \), so:
\[
\text{Horizontal shift} = -\frac{-\frac{5\pi}{24}}{\frac{5}{6}} = \frac{5\pi}{24} \times \frac{6}{5} = \frac{\pi}{4}
\]
Thus, the horizontal shift is \( \frac{\pi}{4} \) units to the right.

### Summary:
- **Amplitude**: \( 2 \)
- **Period**: \( \frac{12\pi}{5} \)
- **Horizontal shift**: \( \frac{\pi}{4} \) units to the right

Would you like more details or have any questions?

#### 5 Related Questions:
1. How do changes in amplitude affect the graph of a sine function?
2. What happens to the period if \( B \) is increased in a sine function?
3. How does the horizontal shift affect the starting point of the sine curve?
4. What is the phase shift for a function \( y = \sin \left( 2x - \frac{\pi}{6} \right) \)?
5. How would the graph of \( y = 2 \sin \left( \frac{5}{6}x \right) \) compare to the given function?

#### Tip:
The period of any sine function can be found by dividing \( 2\pi \) by the coefficient of \( x \), so always check the term in front of \( x \) first!

Find the amplitude, period, and horizontal shift of the function y = 2 sin(5/6 x - 5π/24).

Learn how to determine the amplitude, period, and horizontal shift of the trigonometric function y = 2 sin(5/6 x - 5π/24). This problem involves key trigonometry concepts like amplitude, period calculation, and phase shift for high school students in Grades 10-12.

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