Final answer:
The focus of the given parabola is (0, 4), the directrix is y = -4, and the axis of symmetry is x = 0.
Step-by-step explanation:
The given function is f(x) = (1/16)x². To identify the focus, directrix, and axis of symmetry, we need to rewrite the equation in the standard form of a parabola. The standard form is y = (1/4f)(x - h)² + k, where (h, k) is the vertex and f is the focal length. Comparing this with the given equation, we can see that h = 0 and k = 0. Therefore, the vertex is at the origin (0, 0). Since the coefficient of x² is positive, the parabola opens upwards. This means that the focus and directrix are above the vertex.
To find the focus, we use the formula f = 1/(4a), where a is the coefficient of x². In this case, a = 1/16. So, f = 1/(4(1/16)) = 4. The focus is 4 units above the vertex, so its coordinates are (0, 4).
The directrix is a horizontal line that is equidistant from the vertex as the focus. Since the parabola opens upwards, the directrix is below the vertex. Its equation is given by y = -f. Substituting the value of f, we get y = -4. Therefore, the directrix is the horizontal line y = -4.
The axis of symmetry is a vertical line that passes through the vertex and is perpendicular to the directrix. In this case, the axis of symmetry is the y-axis (x = 0).