Question
Periodic Functions Online Practice Part 2
Complete this assessment to review what you’ve learned. It will not count toward your grade.
9 of 119 of 11 Items
Question
Rewrite the cosine function f(x)=−3cos(x−π8)+2
as a translated sine function.(1 point)
Responses
f(x)=−3sin(x+π4)+2
f left parenthesis x right parenthesis equals negative 3 sine left parenthesis x plus Start Fraction pi over 4 End Fraction right parenthesis plus 2
f(x)=−3sin(x−5π8)+2
f left parenthesis x right parenthesis equals negative 3 sine left parenthesis x minus Start Fraction 5 pi over 8 End Fraction right parenthesis plus 2
f(x)=−3sin(x+3π8)+2
f left parenthesis x right parenthesis equals negative 3 sine left parenthesis x plus Start Fraction 3 pi over 8 End Fraction right parenthesis plus 2
f(x)=−3sin(x+π24)+2
f left parenthesis x right parenthesis equals negative 3 sine left parenthesis x plus Start Fraction pi over 24 End Fraction right parenthesis plus 2
Remaining Attempts : 3
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Answers
GPT-4o mini
To rewrite the cosine function \( f(x) = -3 \cos(x - \frac{\pi}{8}) + 2 \) as a translated sine function, we can use the identity that relates sine and cosine:
\[
\cos(x) = \sin\left(x + \frac{\pi}{2}\right)
\]
We will also need to account for the amplitude and translation.
1. Start with the cosine function:
\[
f(x) = -3 \cos(x - \frac{\pi}{8}) + 2
\]
2. Rewrite the cosine in terms of sine:
\[
f(x) = -3 \sin\left(x - \frac{\pi}{8} + \frac{\pi}{2}\right) + 2
\]
3. Simplify the inside of the sine:
\[
x - \frac{\pi}{8} + \frac{\pi}{2} = x - \frac{\pi}{8} + \frac{4\pi}{8} = x + \frac{3\pi}{8}
\]
4. Now putting it into the function, we have:
\[
f(x) = -3 \sin\left(x + \frac{3\pi}{8}\right) + 2
\]
5. Since there is a negative sign in front of the sine, we can rewrite it:
\[
f(x) = -3 \sin\left(x + \frac{3\pi}{8}\right) + 2 = -3 \sin\left(x - \left(-\frac{3\pi}{8}\right)\right) + 2
\]
Now we can look at the provided choices to find the correct answer:
- **Correct Answer: \( f(x) = -3 \sin\left(x + \frac{3\pi}{8}\right) + 2 \)**
So the correct option is:
\[
\boxed{f(x) = -3\sin(x + \frac{3\pi}{8}) + 2}
\]
\[
\cos(x) = \sin\left(x + \frac{\pi}{2}\right)
\]
We will also need to account for the amplitude and translation.
1. Start with the cosine function:
\[
f(x) = -3 \cos(x - \frac{\pi}{8}) + 2
\]
2. Rewrite the cosine in terms of sine:
\[
f(x) = -3 \sin\left(x - \frac{\pi}{8} + \frac{\pi}{2}\right) + 2
\]
3. Simplify the inside of the sine:
\[
x - \frac{\pi}{8} + \frac{\pi}{2} = x - \frac{\pi}{8} + \frac{4\pi}{8} = x + \frac{3\pi}{8}
\]
4. Now putting it into the function, we have:
\[
f(x) = -3 \sin\left(x + \frac{3\pi}{8}\right) + 2
\]
5. Since there is a negative sign in front of the sine, we can rewrite it:
\[
f(x) = -3 \sin\left(x + \frac{3\pi}{8}\right) + 2 = -3 \sin\left(x - \left(-\frac{3\pi}{8}\right)\right) + 2
\]
Now we can look at the provided choices to find the correct answer:
- **Correct Answer: \( f(x) = -3 \sin\left(x + \frac{3\pi}{8}\right) + 2 \)**
So the correct option is:
\[
\boxed{f(x) = -3\sin(x + \frac{3\pi}{8}) + 2}
\]