The graph of one complete period of a sine curve is given.
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(a) Find the amplitude, period, and horizontal shift. (Assume the absolute value of the horizontal shift is less than the period.)
amplitude    	
7
 


(b) Write an equation that represents the curve in the form
y = a sin(k(x − b)).


To solve this problem, we'll start by analyzing the given information and the general form of a sine function. The equation of a sine function in the form \( y = a \sin(k(x - b)) \) can be described as follows:

- **Amplitude** \( a \): The amplitude is the distance from the midline to the maximum or minimum value. It’s given as \( 7 \).

- **Period** \( T \): The period is the length of one complete cycle of the sine curve. The period is related to the value \( k \) by the formula \( T = \frac{2\pi}{k} \).

- **Horizontal Shift** \( b \): The horizontal shift is the amount the graph is shifted to the left or right from the origin. This shift corresponds to \( b \) in the equation.

Let's go step by step:

### (a) Determine the Period and Horizontal Shift

1. **Amplitude**: It's already given as \( 7 \).

2. **Period**: To find \( k \), we use the period \( T \). Since the period is the length of one complete cycle, we need to identify this from the graph. The period \( T \) is the difference between the x-values where the sine function completes one cycle.

   Let's assume the period \( T \) is given or can be determined from the graph. If \( T \) is the period, then:
   \[
   k = \frac{2\pi}{T}
   \]

3. **Horizontal Shift**: The horizontal shift \( b \) can be determined by identifying where the sine curve starts its cycle. If the graph is shifted to the right by \( b \) units, then \( b \) is positive; if shifted to the left, \( b \) is negative.

### (b) Write the Equation

Once we have \( a \), \( k \), and \( b \), we can write the equation in the form:
\[
y = a \sin(k(x - b))
\]

Let’s fill in the values:

- \( a = 7 \)
- \( k \) can be calculated using \( k = \frac{2\pi}{T} \)
- \( b \) is the horizontal shift.

The final equation would be:
\[
y = 7 \sin\left(k(x - b)\right)
\]

### Example:

- Suppose the period \( T = 4 \), then \( k = \frac{2\pi}{4} = \frac{\pi}{2} \).
- If the horizontal shift \( b = 1 \), then the equation becomes:
  \[
  y = 7 \sin\left(\frac{\pi}{2}(x - 1)\right)
  \]

If you provide the period and the horizontal shift based on the graph, I can give you the exact equation.

Would you like more details or further assistance with specific values? Here are some related questions:

1. How do you determine the period of a sine function from its graph?
2. What is the relationship between the amplitude and the vertical stretch of a sine curve?
3. How do you determine the horizontal shift of a sine curve from its graph?
4. What changes in the sine function equation when the graph is reflected across the x-axis?
5. How does adding a vertical shift affect the equation of a sine curve?

**Tip**: When analyzing the graph of a sine function, always check where the curve intersects the midline (y = 0) to help determine the phase shift.

The graph of one complete period of a sine curve is given. (a) Find the amplitude, period, and horizontal shift. (b) Write an equation that represents the curve in the form y = a sin(k(x − b)).

Explore how to find the amplitude, period, and horizontal shift of a sine curve given its graph. This trigonometry problem provides a step-by-step guide to understanding sine function transformations. Ideal for students in Grades 10-12.

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