Final answer:
The vertex of the function f(x) = x² - 13 is at the point (0, -13), and the axis of symmetry is the line x = 0, which is the y-axis.
Step-by-step explanation:
The equation given is f(x) = x² - 13, which is a quadratic function in the form f(x) = ax² + bx + c. To find the vertex of the function, we can use the fact that the vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Because there is no x term in the original equation, it is clear that h = 0. Thus, the vertex is at x = 0. Substituting x = 0 into the equation f(x) results in f(0) = 0² - 13 = -13. Therefore, the vertex is at (0, -13).
The axis of symmetry for a parabola is a vertical line that passes through the vertex. Since our vertex's x-coordinate is 0, the axis of symmetry is the line x = 0, which is also the y-axis. This axis is also known as the vertical axis of symmetry.