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). y^2 = - 4x

Verified step by step guidance

1. Recognize that the given equation is $y^2=\; -4x$, which represents a parabola. Since the variable $y$ is squared, this is a parabola that opens either left or right.
2. Rewrite the equation in the standard form of a horizontal parabola: $(y\; -\; k)^2\; =\; 4p(x\; -\; h)$. Here, the vertex is at $(h,\; k)$. In this case, the equation is $y^2\; =\; -4x$, so the vertex is at the origin $(0,\; 0)$ and $4p\; =\; -4$.
3. Solve for $p$ by dividing both sides by 4: $p\; =\; \backslash frac\{-4\}\{4\}\; =\; -1$. The value of $p$ tells us the distance from the vertex to the focus and from the vertex to the directrix. Since $p$ is negative, the parabola opens to the left.
4. Find the focus using the vertex and $p$. For a horizontal parabola opening left or right, the focus is at $(h\; +\; p,\; k)$. Here, the focus is at $(0\; +\; (-1),\; 0)\; =\; (-1,\; 0)$.
5. Find the directrix, which is a vertical line given by $x\; =\; h\; -\; p$. Substitute the values to get $x\; =\; 0\; -\; (-1)\; =\; 1$. So, the directrix is the line $x\; =\; 1$.

Key Concepts

Standard Form of a Parabola

The standard form of a parabola's equation helps identify its orientation and key features. For a parabola that opens left or right, the equation is typically y² = 4px or y² = -4px, where p represents the distance from the vertex to the focus. Recognizing this form allows you to determine the parabola's shape and direction.

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola used to define it, and the directrix is a line perpendicular to the axis of symmetry. For the equation y² = -4x, the focus lies at (p, 0) and the directrix is the vertical line x = -p, where p is derived from the coefficient in the equation. These elements are essential for graphing and understanding the parabola's properties.

Graph Matching Using Parabola Features

Matching an equation to a graph involves analyzing the parabola's orientation, vertex, focus, and directrix. By calculating the focus and directrix from the equation, you can compare these features to the given graphs (a)–(d) to find the correct match. This process reinforces understanding of how algebraic equations translate into geometric shapes.