Questions: The graph of a sine function is shown. Move numbers into the blanks of the sine equation to create the function that represents the graph. f(x)= sin ( x)- 0.5 1 1.5 2 3 π 2 π

The graph of a sine function is shown.

Move numbers into the blanks of the sine equation to create the function that represents the graph.
f(x)=
sin ( x)-
0.5
1
1.5
2
3
π
2 π

Transcript text: The graph of a sine function is shown. Move numbers into the blanks of the sine equation to create the function that represents the graph. \[ f(x)= \] $\qquad$ $\sin ($ $\qquad$ $x)-$ $\qquad$ 0.5 1 1.5 2 3 $\pi$ $2 \pi$

Solution

Solution Steps

Step 1: Identify the amplitude

The amplitude of the sine function is the maximum value of the function. From the graph, the maximum value is 3 and the minimum value is -3. Therefore, the amplitude $ A $ is 3.

Step 2: Determine the period

The period of the sine function is the length of one complete cycle. From the graph, one complete cycle occurs from $ -\frac{\pi}{4} $ to $ \frac{7\pi}{4} $, which is a length of $ \frac{7\pi}{4} - (-\frac{\pi}{4}) = 2\pi $. Therefore, the period $ T $ is $ 2\pi $.

Step 3: Calculate the frequency

The frequency $ B $ is related to the period by the formula $ B = \frac{2\pi}{T} $. Since the period $ T $ is $ 2\pi $, the frequency $ B $ is $ \frac{2\pi}{2\pi} = 1 $.

Step 4: Identify the phase shift

The phase shift $ C $ is the horizontal shift of the function. From the graph, the sine function starts at $ -\frac{\pi}{4} $. Therefore, the phase shift $ C $ is $ -\frac{\pi}{4} $.

Step 5: Identify the vertical shift

The vertical shift $ D $ is the vertical displacement of the function. From the graph, the function oscillates around the y-axis, so the vertical shift $ D $ is 0.