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To determineTo determine the equation of the the sine function shown in the equation graph of, the let sine us function analyze shown the in following the features graph of, let the graph’s analyze:

 the1. key ** featuresAmplitude of** the: function  
:

  ### The Step amplitude  is1 the: distance Identify from the the amplitude mid
The sine wave oscilllineates ( between0 \(y in this = - case3)\) to the and \( peaky or =  trough3.\ From). The the graph, the amplitude (\ amplitude( isA \(\))1 is\ the).

 absolute2 value of. the **Period**:  
 maximum   The sine function completes deviation one from full the centerline (\(y cycle = between \(x  = -0\\pi)):
\\[
A =) 3 and \(x =
 \\]

pi\###). Step The period of the sine function  is therefore:
2:   Determine \ the[
 period
  The \ periodtext (\{(Period} = \piT - (-\\))pi is) the = length  of2 one\pi
   \ full cycle of the sine wave.]

3 From. the graph, the sine wave ** starts at \(xVertical Shift =** :0  
\  ), returns to The the same graph point oscill (atescom sympleting one cycle) atmetr \(x = 2\pi\). Thus:
\[
ically aroundT \(y =  = 0\),2\ sopi
 there\ is no]

 vertical shift.

The4 general formula. for ** thePhase period Shift of a** sine: function  
 is  :
 The\[
 graphT starts = at \ \(frac{x2 =\ pi}{0B\}
)\ with]
 thewhere sine \( curveB inverted\) ( is thegoing frequency. downward Sol),ving which for \( suggestsB a phase\):
 shift\ and[
 reflectionB over = the \frac x{-axis2.\ 

pi}{ThusT,} the = equation \ isfrac:
{\2[
y =\ -\sin(x)
pi\}{2\pi]

}### = Let me know if you  would1 like
\ me]

 to### Step 3: Determine the clarify phase shift any
 partThe of graph this has explanation no!  
Here horizontal are shift five ( relatedit questions for further starts at exploration:
1. What \(x changes in the equation if = the  amplitude were 0\2) instead when of  \(1y?
 =2 . How do0 you\ determine)), the so phase shift the phase of a sine function shift (\(?
3C\.)) is:
\[
C = 0 What is
\]

### the Step 4 equation: Determine the vertical shift of
The sine wave the oscillates symmetrically cosine around the function that \( matchesx this\ graph)-axis (\(?
y = 04\)), so. How does adding a the vertical shift affect vertical the graph of the sine function?
 shift5 (\(D\)) is. Can we rewrite the:
 equation in terms\ of[
 \(Dy = =  a \0cos(bx
\ + c) +]

### Step  d5\)?

:** CombineTip the** information: into The the sine general function sine's equation properties
 (Theam generalplitude, form of period, the phase sine shift function, is and:
 vertical\ shift[
)y are = easy A to verify \ ifsin you(B identifyx the - mid Cline) and + peaks D!
\]
Substituting the values \(A = 3\), \(B = 1\), \(C = 0\), and \(D = 0\), we get:
\[
y = 3\sin(x)
\]

### Final Answer:
The equation of the sine function is:
\[
y = 3\sin(x)
\]

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Do you want a detailed explanation of any specific part? Let me know!

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Here are 5 follow-up questions related to sine functions:
1. What happens to the graph if the amplitude is doubled?
2. How do you find the phase shift if the graph starts at a different \(x\)-coordinate?
3. What is the effect of adding a vertical shift to the sine function?
4. How does changing the frequency (\(B\)) alter the period of the graph?
5. What would the equation look like if this graph was a cosine function instead?

**Tip:** Always analyze the amplitude, period, and shifts separately to easily identify the sine or cosine function's equation.

Determine the equation of the sine function shown in the graph.

Learn how to determine the equation of a sine function from its graph. This problem explores key concepts in trigonometry, including amplitude, period, phase shift, and vertical shift. Step-by-step guidance for students in Grades 9-11.

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