### Instruction: For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$?### Response: I need to find the vertex of the parabola first, since that is a fixed point.
To find the vertex, I can use the formula $x = -b/2a$ for the $x$-coordinate, where $b = 0$ and $a = 1$ in this case.
So the vertex has $x$-coordinate $0$, and to find the $y$-coordinate, I plug in $x = 0$ into the parabola equation.
I get $y = 0^2 + a^2 = a^2$.
So the vertex is $(0, a^2)$.
Now I want to find the values of $a$ that make the line $y = x + a$ pass through this point.
To do that, I plug in $x = 0$ and $y = a^2$ into the line equation and see what values of $a$ make it true.
I get $a^2 = 0 + a$, which simplifies to $a^2 - a = 0$.
I can factor this as $a(a-1) = 0$.
So the solutions are $a = 0$ or $a = 1$.
These are the only two values of $a$ that make the line pass through the vertex of the parabola.