To write the equation of a sine function in the form $y = a \sin(k(x - b))$, we need to identify three key characteristics of the curve:

1. Amplitude $a$
2. Period $T$
3. Horizontal shift $b$

Here’s the equation structure:

$y = a \sin(k(x - b))$

Where:

• $a$ is the amplitude.
• $k$ is related to the period by the formula $T = \frac{2\pi}{k}$.
• $b$ is the horizontal shift.

Since your amplitude is $7$, we can substitute $a = 7$.

Assuming the correct value of the period $T$ has been provided or verified, we can calculate $k$. Let’s say the period is $T$, then:

$k = \frac{2\pi}{T}$

Finally, the horizontal shift $b$ has also been confirmed as correct.

Putting it all together, the equation becomes:

$y = 7 \sin(k(x - b))$

Substitute the correct value of $k$ and $b$ from your known parameters to complete the equation.

Would you like to provide the period or horizontal shift values to further simplify the equation?

Follow-up Questions:

1. How do you calculate the value of $k$ from the period?
2. Can you explain the effect of amplitude $a$ on the graph?
3. How does the horizontal shift $b$ change the position of the graph?
4. How would you determine the period of the sine function from its graph?
5. What is the difference between a sine and cosine graph in terms of horizontal shift?

Tip:

Remember that increasing the amplitude stretches the graph vertically, while decreasing it compresses it.