The function \( f(x) = x^2 + 19 \) is a quadratic function in the standard form \( f(x) = ax^2 + bx + c \), where \( a = 1 \), \( b = 0 \), and \( c = 19 \).
To find the vertex, we use the formula for the x-coordinate of the vertex, which is given by:
\[ x = -\frac{b}{2a} \]
Substituting \( a \) and \( b \) into the formula:
\[ x = -\frac{0}{2 \cdot 1} = 0 \]
Next, we find the y-coordinate of the vertex by plugging this x-coordinate back into the function \( f(x) \):
\[ f(0) = (0)^2 + 19 = 19 \]
Thus, the vertex of the function is:
\[ (0, 19) \]
The axis of symmetry is the vertical line that goes through the vertex. Since the x-coordinate of the vertex is 0, the equation of the axis of symmetry is:
\[ x = 0 \]
So, the vertex of the function is (0, 19).