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. Identify the form of the given parabola equation. The equation is $x^2=4y$, which matches the standard form of a vertical parabola: $x^2=\; 4$py.
2. From the equation, compare $4p$ to the coefficient of $y$. Here, $4p\; =\; 4$, so solve for $p$: $p\; =\; 1$.
3. Recall that for a parabola in the form $x^2=\; 4$py, the vertex is at the origin $(0,0)$, the focus is at $(0,\; p)$, and the directrix is the line $y\; =\; -p$.
4. Using the value of $p\; =\; 1$, write the coordinates of the focus as $(0,\; 1)$ and the equation of the directrix as $y\; =\; -1$.
5. To match the equation to one of the graphs (a)–(d), look for the graph showing a parabola opening upwards with vertex at the origin, focus at $(0,\; 1)$, and directrix $y\; =\; -1$.

Key Concepts

Standard Form of a Parabola

The standard form of a parabola with a vertical axis is x² = 4py, where p represents the distance from the vertex to the focus. Understanding this form helps identify key features like the focus and directrix by comparing the given equation to the standard form.

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola, and the directrix is a line outside it, both equidistant from the vertex. For x² = 4py, the focus is at (0, p) and the directrix is the line y = -p. These elements define the parabola's shape and position.

Graph Matching Using Parabola Features

Matching equations to graphs requires identifying the parabola's orientation, vertex, focus, and directrix. By calculating these from the equation, you can compare them to the labeled graphs to find the correct match based on shape and position.