amplitude
7

Correct: Your answer is correct.
period

Correct: Your answer is correct.
horizontal shift

Correct: Your answer is correct.

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

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.

Write an equation that represents the curve in the form y = a sin(k(x − b)) with the given amplitude, period, and horizontal shift.

Learn how to write the equation of a sine function in the form y = a sin(k(x − b)) by identifying amplitude, period, and horizontal shift. This problem covers key trigonometric concepts and is suitable for students in Grades 10-12.

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