Finding the Translated Parabola: y=x² with Root at x=4

Which parabola is the translation of the graph of the function $y=x^2$

and is positive in all areas except $x=4$?

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

• Step 1: Identify the desired vertex of the parabola.
• Step 2: Use the vertex form of the parabola, $y = (x - h)^2 + k$.
• Step 3: Verify the solution with problem constraints.

Now, let's work through each step:
Step 1: We need the parabola to have a vertex such that it equals zero at $x = 4$. Thus, the vertex is $(4, 0)$.
Step 2: Using the vertex form, substitute $h = 4$ and $k = 0$ into $y = (x - h)^2 + k$, resulting in $y = (x - 4)^2$. This equation ensures that $y = 0$ only when $x = 4$.
Step 3: This parabola is positive for values of $x$ other than 4, as the square of any nonzero number is positive. Thus, it meets the specified condition of being positive except at $x = 4$.

Therefore, the solution to the problem is $y = (x - 4)^2$.