The curve above is the graph of a sinusoidal function. It goes through the points  
(
−
4
,
0
)
  and  
(
2
,
0
)
 . Write an equation of the form  
f
(
x
)
=
A
sin
(
B
(
x
−
D
)
)
  whose graph matches the given graph.
f
(
x
)
=
  
  

To find the equation of the sinusoidal function \( f(x) = A \sin(B(x - D)) \), let's follow these steps:

### 1. **Determine the period:**
   The sine function has zeros at \( x = -4 \) and \( x = 2 \), meaning these are consecutive zeros of the sine wave. The distance between them is \( 2 - (-4) = 6 \). This distance represents half of a full period of the sine wave, so the full period is:
   \[
   \text{Period} = 2 \times 6 = 12
   \]

### 2. **Find \( B \):**
   The period of a sine function is related to \( B \) by the formula:
   \[
   \text{Period} = \frac{2\pi}{B}
   \]
   Substituting the known period:
   \[
   12 = \frac{2\pi}{B}
   \]
   Solving for \( B \):
   \[
   B = \frac{2\pi}{12} = \frac{\pi}{6}
   \]

### 3. **Determine \( D \) (Phase Shift):**
   Since the sine function typically starts at \( x = 0 \), but here the sine wave has a zero at \( x = -4 \), the function has been shifted horizontally. This shift is \( D \), and it corresponds to the first zero point at \( x = -4 \). Therefore, \( D = -4 \).

### 4. **Determine \( A \) (Amplitude):**
   From the problem, we know the sine function passes through \( (x = -4, y = 0) \) and \( (x = 2, y = 0) \), but we don't have explicit information about the maximum or minimum values of the sine function. However, assuming the standard sine function shape, we can set the amplitude \( A = 1 \), unless other information is provided to suggest otherwise.

### Final Equation:
The equation of the sinusoidal function is:
\[
f(x) = \sin\left(\frac{\pi}{6}(x + 4)\right)
\]

Would you like more details on any part of this process?

---

Here are 5 questions to extend this topic:
1. How does changing the value of \( A \) affect the graph of the sinusoidal function?
2. What happens to the graph if you change the value of \( B \)?
3. How can we modify the equation if the sine function had a maximum at \( x = -4 \)?
4. What would the equation be if the curve was a cosine function instead of a sine function?
5. How can you determine the vertical shift of a sinusoidal function?

**Tip:** The period of any sinusoidal function can be easily calculated using the formula \( \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( x \) inside the sine or cosine function.

The curve above is the graph of a sinusoidal function. It goes through the points (-4, 0) and (2, 0). Write an equation of the form f(x) = A sin(B(x − D)) whose graph matches the given graph.

Learn how to write the equation of a sinusoidal function passing through the points (-4, 0) and (2, 0). This trigonometry problem involves finding the amplitude, period, and phase shift of the sine function. Suitable for high school students in Grades 10-12.

Math Problem Statement

The curve above is the graph of a sinusoidal function. It goes through the points
( − 4 , 0 ) and
( 2 , 0 ) . Write an equation of the form
f ( x )

A sin ( B ( x − D ) ) whose graph matches the given graph. f ( x )

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation