Question

Use the table to answer the question.

θ f(θ)

0
π4
π2
3π4
π
Use the table of values to graph the f(θ)=4sin(2θ) and find the amplitude, midline, and period.

(1 point)
Responses

amplitude=2, midline: y=0, period=π4
amplitude=2, midline: y equals 2 , period equals Start Fraction pi over 4 End Fraction

amplitude=4, midline: y=0, period=π
amplitude=4, midline: y equals 0 , period equals pi

amplitude=4, midline: y=2, period=π4
amplitude=4, midline: y equals 3 , period equals pi

amplitude=2, midline: y=4, period=π2

Which option is correct

Answer 1

To analyze the function \( f(\theta) = 4 \sin(2\theta) \), we need to find the amplitude, midline, and period.

1. **Amplitude**:
The amplitude of a sine function \( A \sin(B\theta) \) is given by the absolute value of \( A \). Here, \( A = 4 \), so the amplitude is:
\[
\text{Amplitude} = 4
\]

2. **Midline**:
The midline of a sine function is defined by the vertical shift of the graph, which is typically represented by \( D \) in the general form \( A \sin(B\theta) + D \). In this case, there is no vertical shift (i.e., \( D = 0 \)), so:
\[
\text{Midline} = y = 0
\]

3. **Period**:
The period of a sine function \( A \sin(B\theta) \) is calculated by the formula:
\[
\text{Period} = \frac{2\pi}{|B|}
\]
where \( B \) is the coefficient of \( \theta \). Here \( B = 2 \), so:
\[
\text{Period} = \frac{2\pi}{2} = \pi
\]

Based on this analysis, the correct values for the function \( f(\theta) = 4 \sin(2\theta) \) are:
- Amplitude: 4
- Midline: \( y = 0 \)
- Period: \( \pi \)

Hence, the correct option from the responses is:
- **amplitude=4, midline: y=0, period=π**.

Answer 2

Are you sure

Answer 3

Let's verify the components for the function \( f(\theta) = 4 \sin(2\theta) \) step-by-step again:

1. **Amplitude**:
- The amplitude of a sine function in the form \( A \sin(B\theta) \) is given by \( |A| \). For \( f(\theta) = 4 \sin(2\theta) \), \( A = 4 \).
- Therefore, the **amplitude** is \( 4 \).

2. **Midline**:
- The midline of the sine function is determined by the vertical shift, represented by \( D \) in the formula \( f(\theta) = A \sin(B\theta) + D \). For \( f(\theta) = 4 \sin(2\theta) \), there is no \( D \), so \( D = 0 \).
- Therefore, the **midline** is \( y = 0 \).

3. **Period**:
- The period of a sine function is given by the formula \( \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( \theta \). Here, \( B = 2 \).
- Therefore, the **period** is:
\[
\text{Period} = \frac{2\pi}{2} = \pi
\]

Putting it all together, we have:
- Amplitude: \( 4 \)
- Midline: \( y = 0 \)
- Period: \( \pi \)

Thus, the correct option is indeed:
- **amplitude=4, midline: y=0, period=π**.

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