The given function is $y = 3x^2 - 4$, and it is translated by the vector $\begin{pmatrix} 0 \\ 3 \end{pmatrix}$. This represents a vertical translation, where the graph is shifted upward by 3 units.

When a function is translated vertically by $k$ units, the transformation applied to the function is: $y = f(x) + k$ In this case, $k = 3$, so we apply the transformation: $y = (3x^2 - 4) + 3 = 3x^2 - 4 + 3 = 3x^2 - 1$

Thus, the resulting function after translation is: $y = 3x^2 - 1$

From the provided options, the correct answer is:

• Option a. $y = 3x^2 - 1$

Would you like further details or explanations?

Here are 5 related questions:

1. How would the function change if translated by $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$?
2. What is the effect of a horizontal translation on a quadratic function?
3. How do you find the vertex of a translated quadratic function?
4. How can we graphically represent a translation of a quadratic function?
5. How does vertical stretching affect a quadratic function?

Tip: A vertical translation only affects the constant term in the function, moving the entire graph up or down.