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).
a. b. c. d.
y² = 4x

Verified step by step guidance

1. Recognize that the given equation \(y^2 = 4x\) is a parabola in the form \(y^2 = 4px\), where the parabola opens horizontally because the \(y\) variable is squared.
2. Identify the value of \(p\) by comparing \(y^2 = 4x\) to the standard form \(y^2 = 4px\). Here, \$4p = 4\(, so \)p = 1$.
3. Recall that for a parabola \(y^2 = 4px\) that opens to the right, the focus is located at \((p, 0)\) and the directrix is the vertical line \(x = -p\).
4. Using \(p = 1\), write the coordinates of the focus as \((1, 0)\) and the equation of the directrix as \(x = -1\).
5. Match the parabola with the graph that shows a parabola opening to the right with focus at \((1, 0)\) and directrix \(x = -1\).

Key Concepts

Standard Form of a Parabola

The standard form of a parabola's equation helps identify its orientation and key features. For example, y² = 4ax represents a parabola opening right or left, where 'a' determines the distance from the vertex to the focus and directrix. Recognizing this form is essential to find the focus and directrix.

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola, and the directrix is a fixed line outside it. Every point on the parabola is equidistant from the focus and the directrix. For y² = 4ax, the focus is at (a, 0) and the directrix is the vertical line x = -a.

Graphing Parabolas and Matching Equations

Understanding how the equation relates to the graph allows matching equations to their corresponding parabolas. The orientation (horizontal or vertical), vertex location, and distances to focus and directrix guide the sketching and identification of the correct graph among options.