To determine the required transformation Jasmine needs to apply, we must analyze the given function and the changes made:

1. The original function is $f(x) = -\sin\left( 2x + \frac{\pi}{4} \right) - 1$.
2. Jasmine's first step is rewriting it as $f(x) = -\sin\left( 2\left(x + \frac{\pi}{8} \right)\right) - 1$.

To convert a sine function to a cosine function, we recognize that a cosine function is essentially a sine function shifted by $\frac{\pi}{2}$. In this case, the argument of the sine function is $2x + \frac{\pi}{8}$, and we need to shift the sine function horizontally by an appropriate amount to make it match the cosine form.

Upon comparing the sine and cosine forms:

• The argument of the sine function in Jasmine's equation is $2\left(x + \frac{\pi}{8}\right)$.
• The argument of the cosine function should be $2\left(x - \frac{\pi}{8}\right)$, which suggests the shift is to the right.

Therefore, the correct transformation is to shift the graph $\frac{\pi}{8}$ units to the right.

Answer: Option #3. Jasmine needs to shift the graph an additional $\frac{\pi}{4}$ units to the right to obtain the cosine function $f(x) = -\cos\left( 2\left(x - \frac{\pi}{8} \right)\right) - 1$.

Would you like further clarification or additional steps? Here are some related questions for further exploration:

1. How does a phase shift affect the graph of a sine or cosine function?
2. What is the general formula for converting a sine function to a cosine function?
3. How do amplitude and vertical shifts impact the graph of trigonometric functions?
4. What is the difference between a horizontal shift and a phase shift in trigonometric functions?
5. How do the coefficients in front of $x$ and within the parentheses affect the period of sine and cosine functions?

Tip: When converting between sine and cosine functions, always remember that the phase shift represents a horizontal translation and is related to the argument of the trigonometric function.