Find an equation for a sinusoidal function that has a period 2pi, amplitude 3, and contains the point (-2pi,1)

To find an equation for a sinusoidal function with a given period, amplitude, and a specific point, we can use the general form of a sinusoidal function:

\[
y = a \sin(bx - c) + d \quad \text{or} \quad y = a \cos(bx - c) + d
\]

where:
- \( a \) is the amplitude (the maximum distance from the midline),
- \( b \) affects the period (the length of one complete cycle),
- \( c \) is the phase shift (horizontal shift), and
- \( d \) is the vertical shift (midline of the sinusoid).

**Given Information:**
1. Period \( = 2\pi \)
2. Amplitude \( = 3 \)
3. Contains the point \( (-2\pi, 1) \)

### Step 1: Determine the amplitude (\(a\))

The amplitude is given as 3, so:

\[
a = 3
\]

### Step 2: Determine the period to find \(b\)

The period of a sine or cosine function is given by:

\[
\text{Period} = \frac{2\pi}{b}
\]

We know the period is \(2\pi\):

\[
2\pi = \frac{2\pi}{b}
\]

Solving for \(b\):

\[
b = 1
\]

### Step 3: Determine the vertical shift (\(d\))

To find \(d\), we need more information about the sinusoid's midline. However, if we assume there is no vertical shift (since only one point is given and we are not told otherwise), we can initially set \(d = 0\). We'll confirm this later.

### Step 4: Determine the phase shift (\(c\))

We use the point \((-2\pi, 1)\) to find the phase shift. 

Assuming a sine function:

\[
y = 3 \sin(x - c)
\]

Using the point \((-2\pi, 1)\):

\[
1 = 3 \sin(-2\pi - c)
\]

\[
\sin(-2\pi - c) = \frac{1}{3}
\]

Since \(\sin(-2\pi) = 0\), we must have:

\[
\sin(-c) = \frac{1}{3}
\]

Thus:

\[
-c = \arcsin\left(\frac{1}{3}\right)
\]

\[
c = -\arcsin\left(\frac{1}{3}\right)
\]

### Step 5: Construct the function

The equation of the sinusoidal function becomes:

\[
y = 3 \sin\left(x + \arcsin\left(\frac{1}{3}\right)\right)
\]

Alternatively, if using cosine (which is shifted differently):

\[
y = 3 \cos\left(x - \arccos\left(\frac{1}{3}\right)\right)
\]

**To summarize:**

1. **Sinusoidal Function using Sine**:
   \[
   y = 3 \sin\left(x + \arcsin\left(\frac{1}{3}\right)\right)
   \]

2. **Sinusoidal Function using Cosine**:
   \[
   y = 3 \cos\left(x - \arccos\left(\frac{1}{3}\right)\right)
   \]

Both forms fit the conditions, depending on the chosen phase shift direction.

Would you like further details or have any questions?

Here are some related questions:

1. What is the effect of changing the amplitude in a sinusoidal function?
2. How do you determine the phase shift of a sinusoidal function?
3. What happens if the period of a sinusoidal function is different from \(2\pi\)?
4. How can we confirm if there is a vertical shift in the sinusoidal function?
5. How would the function change if the point given was different?

**Tip:** Knowing the unit circle values can greatly help in solving problems involving trigonometric functions and their transformations.

Discover how to find an equation for a sinusoidal function with a period of 2π, amplitude of 3, and containing the point (-2π, 1). Learn step-by-step methods using trigonometric identities and transformations. Suitable for advanced high school students and anyone interested in sinusoidal functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation