https://file.aiquickdraw.com/mathbot/file/1727193786165w3w829ih.jpg

Question content area top left
Part 1
Find an equation for the graph shown.
Question content area bottom left
Part 1
Type the equation of the given graph in the form y equals Upper A sine left parenthesis omega x right parenthesis or y equals Upper A cosine left parenthesis omega x right parenthesis .
yequals

enter your response here
​(Simplify your answer. Type an exact answer in terms of pi. Use integers or fractions for any numbers in the​ expression.)

To determine the equation of the sinusoidal graph shown, we will follow these steps:

### Step 1: Identify the general form
We aim to find the equation in the form of either:
- \( y = A \sin(\omega x) \) or
- \( y = A \cos(\omega x) \)

Where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency, related to the period by \( \omega = \frac{2\pi}{T} \), where \( T \) is the period.

### Step 2: Determine amplitude \( A \)
From the graph, we can observe that the maximum value of \( y \) is 10 and the minimum value is -10. The amplitude \( A \) is given by half the distance between the maximum and minimum values.

\[
A = \frac{10 - (-10)}{2} = 10
\]

Thus, \( A = 10 \).

### Step 3: Determine the period \( T \)
The period is the horizontal distance required for the graph to complete one full cycle. Looking at the graph, the curve repeats every 6 units (from \( x = -1 \) to \( x = 5 \)).

Thus, the period \( T = 6 \).

### Step 4: Find angular frequency \( \omega \)
The angular frequency \( \omega \) is related to the period by the formula \( \omega = \frac{2\pi}{T} \).

\[
\omega = \frac{2\pi}{6} = \frac{\pi}{3}
\]

### Step 5: Choose the appropriate sine or cosine function
The graph appears to start at \( y = 0 \) at \( x = -1 \) and reaches its peak at \( y = 10 \) shortly after. This behavior is typical of a sine function with a horizontal shift.

Thus, the equation has the form:
\[
y = A \sin\left(\omega(x - x_0)\right)
\]

where \( x_0 \) is the horizontal shift. Since the sine curve crosses zero at \( x = -1 \), we use \( x_0 = -1 \).

### Step 6: Write the final equation
Substituting the values we have found into the sine equation:

\[
y = 10 \sin\left(\frac{\pi}{3}(x + 1)\right)
\]

This is the equation for the graph shown.

Would you like further details or explanations on any of the steps?

Here are 5 related questions to expand on the concept:
1. How would the equation change if the amplitude were halved?
2. How would the graph change if the period were doubled?
3. Can you explain the significance of the phase shift \(x_0\)?
4. What would the equation be if the graph were a cosine function?
5. How would adding a vertical shift affect the graph?

**Tip:** Understanding how each parameter in the sine or cosine function (amplitude, period, phase shift, vertical shift) transforms the graph will help in quickly identifying equations for more complex sinusoidal graphs.

Find an equation for the graph shown. Type the equation of the given graph in the form y = A*sin(ωx) or y = A*cos(ωx).

Learn how to find the equation of a sinusoidal graph, where the amplitude is 10, and the period is 6 units. This involves understanding key trigonometric concepts such as amplitude, period, and phase shift. Perfect for high school students studying trigonometric transformations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation