Question

the values of a sine function at intervals of π2 have been recorded in the table. Identify the value of the phase shift (c) and write the equation of the function. (1 point) Responses f(x)=sin(x+π) f left parenthesis x right parenthesis equals sine left parenthesis x plus pi right parenthesis f(x)=sin(x−π2) f left parenthesis x right parenthesis equals sine left parenthesis x minus Start Fraction pi over 2 End Fraction right parenthesis f(x)=sinx+π2 f left parenthesis x right parenthesis equals sine x plus Start Fraction pi over 2 End Fraction f(x)=sin(x+π2) f left parenthesis x right parenthesis equals sine left parenthesis x plus Start Fraction pi over 2 End Fraction right parenthesis

Answer 1

To identify the phase shift and write the appropriate equation for the sine function, we need to analyze the options given.

Recall that the general form of a sine function is:

\[ f(x) = a \sin(b(x - c)) + d \]

where:

• \( c \) represents the phase shift,
• \( a \) is the amplitude,
• \( b \) affects the period (period = \( \frac{2\pi}{|b|} \)),
• \( d \) is the vertical shift.

From the responses provided, we can summarize the functions:

1. \( f(x) = \sin(x + \pi) \)
2. \( f(x) = \sin(x - \frac{\pi}{2}) \)
3. \( f(x) = \sin x + \frac{\pi}{2} \) (not a phase shift, this is a vertical shift)
4. \( f(x) = \sin(x + \frac{\pi}{2}) \)

To find the phase shift \( c \):

• Option 1: \( x + \pi \) indicates a phase shift of \( -\pi \).
• Option 2: \( x - \frac{\pi}{2} \) indicates a phase shift of \( \frac{\pi}{2} \).
• Option 3: This option does not apply since it is a vertical shift.
• Option 4: \( x + \frac{\pi}{2} \) indicates a phase shift of \( -\frac{\pi}{2} \).

Among these options, we can deduce that the correct identification of the phase shift corresponds to option 2, which gives a phase shift of \( \frac{\pi}{2} \).

Final Answer:

Therefore, the value of the phase shift \( c \) is \( \frac{\pi}{2} \), and the equation of the function is:

\[ f(x) = \sin\left(x - \frac{\pi}{2}\right) \]