Find an equation for a sinusoidal function that has a period 2(pi), amplitude 1,and contains the point (-pi,0)

To find the equation of a sinusoidal function with the given properties, let's start by considering the general form of a sinusoidal function:

\[
y = A \sin(Bx + C) + D
\]

where:
- \( A \) is the amplitude.
- \( B \) affects the period.
- \( C \) is the phase shift.
- \( D \) is the vertical shift.

Given:
- Amplitude \( A = 1 \).
- Period \( T = 2\pi \), so \( B = 1 \) (since \( T = \frac{2\pi}{B} \)).
- Vertical shift \( D = 0 \) (no mention of any vertical shift).

So, the equation simplifies to:

\[
y = \sin(x + C)
\]

Next, we'll use the given point \((- \pi, 0)\) to determine the phase shift \( C \).

\[
0 = \sin(-\pi + C)
\]

The sine function equals 0 at multiples of \( \pi \), so:

\[
-\pi + C = k\pi \quad \text{for some integer } k
\]

For simplicity, let's choose \( k = -1 \):

\[
-\pi + C = -\pi
\]

This implies:

\[
C = 0
\]

Thus, the equation is:

\[
y = \sin(x)
\]

However, this equation does not satisfy the condition at \((- \pi, 0)\), so let's reconsider another possibility: using the cosine function.

Given the same amplitude and period:

\[
y = \cos(Bx + C)
\]

Given \( A = 1 \), \( B = 1 \), and using the point \((- \pi, 0)\):

\[
0 = \cos(-\pi + C)
\]

The cosine function equals 0 at odd multiples of \( \frac{\pi}{2} \), so:

\[
-\pi + C = \frac{\pi}{2} + k\pi
\]

Let's solve for \( C \) using \( k = 1 \):

\[
-\pi + C = \frac{3\pi}{2}
\]

\[
C = \frac{3\pi}{2} + \pi = \frac{5\pi}{2}
\]

Thus, the cosine-based sinusoidal function is:

\[
y = \cos\left(x - \frac{\pi}{2}\right)
\]

Alternatively, you could write it as:

\[
y = \sin\left(x + \frac{\pi}{2}\right)
\]

This equation satisfies the conditions.

Do you have any further questions or need clarification on any step?

---

Here are some related questions to consider:
1. How do you determine whether to use sine or cosine for a sinusoidal function?
2. What are the phase shifts in sinusoidal functions?
3. How do you graph sinusoidal functions with phase shifts?
4. What is the significance of different values of \( C \) in sinusoidal functions?
5. How can you check if a given point lies on a sinusoidal curve?

**Tip:** When given a specific point, consider both sine and cosine functions, as either might be a better fit depending on the phase shift!

Learn how to find a sinusoidal function with a period of 2π, amplitude 1, and passing through the point (-π, 0). Understand the principles of sinusoidal functions, including amplitude, period, and phase shift. Suitable for advanced high school students and college-level learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation