In Exercises 1–4, find the focus and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). x^2 = - 4y

Verified step by step guidance

1. Step 1: Identify the form of the equation. The given equation is in the form of x^2 = 4py, where p is the distance from the vertex to the focus and the directrix. In this case, the equation is x^2 = -4y, so 4p = -4, which means p = -1.
2. Step 2: Find the focus of the parabola. The focus of a parabola in the form x^2 = 4py is at the point (0, p). So, in this case, the focus is at the point (0, -1).
3. Step 3: Find the directrix of the parabola. The directrix of a parabola in the form x^2 = 4py is the line y = -p. So, in this case, the directrix is the line y = 1.
4. Step 4: Match the equation to one of the graphs. The graph of a parabola in the form x^2 = 4py opens upward if p > 0 and opens downward if p < 0. Since p = -1 in this case, the parabola opens downward. Look for a graph that shows a parabola opening downward, with a focus at (0, -1) and a directrix at y = 1.
5. Step 5: Remember that the vertex of the parabola is the midpoint between the focus and the directrix. In this case, the vertex is at (0, 0). This can also help you match the equation to the correct graph.

Key Concepts

Parabola Definition

A parabola is a symmetric curve formed by the intersection of a cone with a plane parallel to its side. In algebra, parabolas can be represented by quadratic equations, typically in the form y = ax^2 + bx + c or x = ay^2 + by + c. The orientation of the parabola (opening upwards, downwards, left, or right) depends on the coefficients in the equation.

Focus and Directrix

The focus of a parabola is a fixed point located inside the curve, while the directrix is a line outside the curve. The defining property of a parabola is that any point on the curve is equidistant from the focus and the directrix. For the equation x^2 = -4y, the focus and directrix can be derived from the standard form of a parabola.

Standard Form of a Parabola

The standard form of a parabola that opens vertically is given by the equation (x - h)^2 = 4p(y - k), where (h, k) is the vertex, and p is the distance from the vertex to the focus (and also to the directrix). For the equation x^2 = -4y, it can be rewritten to identify the vertex and the value of p, which helps in determining the focus and directrix.