Parabola Translation: Finding y=x² Shift with Exception at x=-3

To solve this problem, we'll follow these steps:

• Step 1: Understand the necessary translation of the parabola.
• Step 2: Apply the translation to the equation of the parabola.

Now, let's work through each step:

Step 1: We are given the function $y = x^2$. The new parabola must be zero at $x = -3$ and positive everywhere else. This means the vertex of the parabola is at $(-3, 0)$.

Step 2: To move the vertex of $y = x^2$ from $(0, 0)$ to $(-3, 0)$, we need a horizontal shift to the left by 3 units. The translation is represented by replacing $x$ with $x + 3$ in the function:

$y = (x + 3)^2$

This new equation reflects a parabola that opens upwards and has its vertex at $(-3, 0)$, which means it is zero only at $x = -3$ and positive everywhere else.

Therefore, the solution to the problem is $y = (x + 3)^2$.