Question
Use the table to answer the question.
x 0 π2
π
3π2
2π
f(x) 1 0 −1
0 1
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)=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 minus Start Fraction pi over 2 End Fraction right parenthesis
f(x)=sin(x+π2)
x 0 π2
π
3π2
2π
f(x) 1 0 −1
0 1
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)=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 minus Start Fraction pi over 2 End Fraction right parenthesis
f(x)=sin(x+π2)
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To identify the phase shift and write the equation of the sine function, we need to analyze the given table and the values provided for \( f(x) \).
The table provides:
- When \( x = 0 \), \( f(0) = 1 \)
- When \( x = \frac{\pi}{2} \), \( f\left(\frac{\pi}{2}\right) = 0 \)
- When \( x = \pi \), \( f(\pi) = -1 \)
- When \( x = \frac{3\pi}{2} \), \( f\left(\frac{3\pi}{2}\right) = 0 \)
- When \( x = 2\pi \), \( f(2\pi) = 1 \)
This behavior suggests that the function is a sine function, which typically has a maximum value of 1 at \( x = 0 \) without any phase shift.
However, the function appears to have values shifted one quarter of the way through its cycle. Specifically, a typical sine function, \( f(x) = \sin x \), would attain its maximum at \( x = 0 \). Since our function is also at \( f(0) = 1 \), this suggests it is a sine function, but we need to determine if there's a phase shift.
From the table, when \( x \) is at its maximum (\( f(x) = 1 \)), it results in a sine function that shifted to the left. The value of \( \frac{\pi}{2} \) would naturally give a 0 value from \( \sin x \), but here, it appears to be shifted to give a max at \( x = 0 \).
Recognizing this, the phase shift can be determined by observing that:
\[ f(x) = \sin(x + \frac{\pi}{2}) \]
Since \( \sin(x + \frac{\pi}{2}) \) would match the given values in the table,
The shifted sine function corresponds to:
- A **phase shift** to the left, which can be represented as "c" in the context of sine functions. Here, \( c = \frac{\pi}{2} \).
Thus, the equation of the function is:
\[ f(x) = \sin(x + \frac{\pi}{2}) \]
The correct response that reflects this fact is:
**f(x) = sin(x + π/2)**.
The table provides:
- When \( x = 0 \), \( f(0) = 1 \)
- When \( x = \frac{\pi}{2} \), \( f\left(\frac{\pi}{2}\right) = 0 \)
- When \( x = \pi \), \( f(\pi) = -1 \)
- When \( x = \frac{3\pi}{2} \), \( f\left(\frac{3\pi}{2}\right) = 0 \)
- When \( x = 2\pi \), \( f(2\pi) = 1 \)
This behavior suggests that the function is a sine function, which typically has a maximum value of 1 at \( x = 0 \) without any phase shift.
However, the function appears to have values shifted one quarter of the way through its cycle. Specifically, a typical sine function, \( f(x) = \sin x \), would attain its maximum at \( x = 0 \). Since our function is also at \( f(0) = 1 \), this suggests it is a sine function, but we need to determine if there's a phase shift.
From the table, when \( x \) is at its maximum (\( f(x) = 1 \)), it results in a sine function that shifted to the left. The value of \( \frac{\pi}{2} \) would naturally give a 0 value from \( \sin x \), but here, it appears to be shifted to give a max at \( x = 0 \).
Recognizing this, the phase shift can be determined by observing that:
\[ f(x) = \sin(x + \frac{\pi}{2}) \]
Since \( \sin(x + \frac{\pi}{2}) \) would match the given values in the table,
The shifted sine function corresponds to:
- A **phase shift** to the left, which can be represented as "c" in the context of sine functions. Here, \( c = \frac{\pi}{2} \).
Thus, the equation of the function is:
\[ f(x) = \sin(x + \frac{\pi}{2}) \]
The correct response that reflects this fact is:
**f(x) = sin(x + π/2)**.
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