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.