Use the table to answer the question.

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

amplitude=2, midline: y=4
, period=π2

Answer 1

To analyze the function \( f(\theta) = 4\sin(2\theta) \), we need to identify its amplitude, midline, and period based on its standard form.

1. Amplitude: The amplitude of a sine function \( A\sin(B\theta) \) is given by the coefficient \( A \). In this case, \( A = 4 \), so the amplitude is \( 4 \).

2. Midline: The midline of a sine function is given by the vertical shift. Since there is no constant added or subtracted from the function (i.e., no \( + C \)), the midline is \( y = 0 \).

3. Period: The period of the sine function is calculated using the formula \( \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( \theta \). For our function, \( B = 2 \). Thus, the period is:

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

Putting this all together, we find:

• Amplitude: \( 4 \)
• Midline: \( y = 0 \)
• Period: \( \pi \)

So, the correct response would be:

amplitude=4, midline: y=0, period=π