Question
Periodic Function Flexibility Practice
Complete this assessment to review what you’ve learned. It will not count toward your grade.
1 of 51 of 5 Items
Question
Jasmine is converting f(x)=−sin(2x+π4)−1 to a cosine function.
Her first step was to rewrite the function as f(x)=−sin(2(x+π8))−1.
What transformation does she need to apply?
Option #1: Jasmine needs to shift the graph an additional π8 units to the right to obtain the cosine function f(x)=−cos(2x)−1.
Option #2: Jasmine needs to shift the graph an additional π4 units to the left to obtain the cosine function f(x)=−cos(2(x+3π8))−1.
Option #3: Jasmine needs to shift the graph an additional π4 units to the right to obtain the cosine function f(x)=−cos(2(x−π8))−1.
(1 point)
Option #
describes the needed transformation.
Remaining Attempts : 3
Skip to navigation
Answers
GPT-4o mini
To convert the sine function \( f(x) = -\sin(2x + \frac{\pi}{4}) - 1 \) to a cosine function, we can use the fact that \( -\sin(x) = \cos(x + \frac{\pi}{2}) \).
First, we rewrite the sine function:
1. Start with \( f(x) = -\sin(2x + \frac{\pi}{4}) - 1 \).
2. Rewrite the sine part: \(-\sin(2x + \frac{\pi}{4}) = \cos\left(2x + \frac{\pi}{4} + \frac{\pi}{2}\right) = \cos\left(2x + \frac{3\pi}{4}\right)\).
Thus, the function can initially be expressed as:
\[ f(x) = \cos\left(2x + \frac{3\pi}{4}\right) - 1 \]
Next, to have it in the form \( f(x) = -\cos(2x) -1 \), we need to manipulate the angle in the cosine:
From \( \cos\left(2x + \frac{3\pi}{4}\right) \) to \( -\cos(2x) \), we consider adjusting the phase:
To convert \( 2x + \frac{3\pi}{4} \) back to a cosine function with a phase shift of \( \frac{\pi}{4} \), we need to shift to the right by \( \frac{3\pi}{4} - 0 = \frac{3\pi}{4} \).
To summarize, the transformation that would lead to \( f(x)= -\cos(2(x - \frac{\pi}{8})) - 1 \) requires shifting:
So, the correct option is:
**Option #3**: Jasmine needs to shift the graph an additional \( \frac{\pi}{4} \) units to the right to obtain the cosine function \( f(x)=-\cos(2(x-\frac{\pi}{8}))-1 \).
First, we rewrite the sine function:
1. Start with \( f(x) = -\sin(2x + \frac{\pi}{4}) - 1 \).
2. Rewrite the sine part: \(-\sin(2x + \frac{\pi}{4}) = \cos\left(2x + \frac{\pi}{4} + \frac{\pi}{2}\right) = \cos\left(2x + \frac{3\pi}{4}\right)\).
Thus, the function can initially be expressed as:
\[ f(x) = \cos\left(2x + \frac{3\pi}{4}\right) - 1 \]
Next, to have it in the form \( f(x) = -\cos(2x) -1 \), we need to manipulate the angle in the cosine:
From \( \cos\left(2x + \frac{3\pi}{4}\right) \) to \( -\cos(2x) \), we consider adjusting the phase:
To convert \( 2x + \frac{3\pi}{4} \) back to a cosine function with a phase shift of \( \frac{\pi}{4} \), we need to shift to the right by \( \frac{3\pi}{4} - 0 = \frac{3\pi}{4} \).
To summarize, the transformation that would lead to \( f(x)= -\cos(2(x - \frac{\pi}{8})) - 1 \) requires shifting:
So, the correct option is:
**Option #3**: Jasmine needs to shift the graph an additional \( \frac{\pi}{4} \) units to the right to obtain the cosine function \( f(x)=-\cos(2(x-\frac{\pi}{8}))-1 \).