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 $x^2 = 4ay$, which is a standard form of a parabola that opens upwards or downwards.
2. Step 2: Identify the value of 'a'. In the given equation, $4a = 4$, so $a = 1$.
3. Step 3: Find the focus of the parabola. For a parabola in the form $x^2 = 4ay$, the focus is at the point $(0, a)$, so in this case, the focus is at the point $(0, 1)$.
4. Step 4: Find the directrix of the parabola. For a parabola in the form $x^2 = 4ay$, the directrix is the line $y = -a$, so in this case, the directrix is the line $y = -1$.
5. Step 5: Match the equation to one of the graphs. Look for a graph of a parabola that opens upwards (since 'a' is positive), with a focus at $(0, 1)$ and a directrix at $y = -1$.

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 of the equation.

Focus and Directrix

The focus of a parabola is a fixed point located inside the curve, while the directrix is a line perpendicular to the axis of symmetry of the parabola. 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 its standard form.

Standard Form of a Parabola

The standard form of a parabola that opens upwards or downwards 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). In 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.