Problem 21 - College Algebra with Modeling and Visualization

Question

Graph the parabola. Label the vertex, focus, and directrix. $$ -4 x=y^{2} $$

Step-by-Step Solution

Answer

Graph the parabola with vertex at (0,0), focus at (-1,0), and directrix x=1.

Step 1: Identify the Standard Form

The given equation is \(-4x = y^2\). This equation is in the form \(y^2 = 4px\), which indicates a parabola that opens either to the left or right. In this standard form, \(p\) represents the distance from the vertex to the focus and from the vertex to the directrix.

Step 2: Find the Vertex

For a parabola that is horizontal (opening left or right), the vertex is at the origin unless otherwise shifted. Here, the vertex is at \((0, 0)\).

Step 3: Determine the Direction and Parameter p

Since we have \(y^2=-4x\), the parabola opens to the left because of the negative sign. Comparing \(-4x = y^2\) to the standard form \(y^2 = 4px\), we equate \(4p = -4\) to get \(p = -1\).

Step 4: Locate the Focus

For a parabola opening to the left, the focus is \(p\) units to the left of the vertex. Thus, the focus is \((-1, 0)\).

Step 5: Find the Directrix

The directrix is a vertical line \(p\) units to the right of the vertex. Because \(p = -1\), the directrix is \(x = 1\).

Step 6: Graph the Parabola

Plot the vertex at \((0, 0)\). Draw the focus at \((-1, 0)\), and the vertical line for the directrix at \(x = 1\). Sketch the parabola opening towards the left, as it curves around the focus.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key feature. It acts as the point where the curve changes direction. Think of it like the tip or the "turning point" of the parabola. In a standard form parabola like the one given, \(-4x = y^2\), the vertex is at the origin unless moved by a transformation.

For the specific equation \(y^2 = -4x\), the vertex is located at \((0, 0)\). It marks the center of symmetry for the parabola. This means the curve looks the same on both sides of this point.

• The vertex gives valuable information about the position of the curve.
• It is crucial for determining the direction in which the parabola opens.

Keeping track of the vertex can make graphing more precise and easier.

Focus of a Parabola

The focus of a parabola is a point from which the distances to points on the curve have a unique property. In our case, the equation given is \(-4x = y^2\), indicating the parabola opens leftwards due to the negative sign. The focus is located using the value \(p\), which represents the distance from the vertex.

For the equation \(-4x = y^2\), we have \(4p = -4\) that leads to \(p = -1\). This means the focus is \(-1\) unit to the left of the vertex \((0, 0)\). Thus the coordinates of the focus are \((-1, 0)\).

• The focus affects the shape and the direction the parabola opens.
• It is always along the axis of symmetry of the parabola.

The focus is vital as it directly influences the curvature of the parabola.

Directrix of a Parabola

The directrix of a parabola is a line opposite the focus relative to the vertex. It helps ensure that the parabola maintains a correct curve shape by providing a reference line.

In our problem \(-4x = y^2\), the directrix is found at a distance \(p\) away, opposite to the direction of the focus. Because \(p = -1\), the directrix is \(x = 1\), a vertical line.

• It lies on the opposite side of the vertex from the focus along the axis of symmetry.
• Every point on the parabola is equidistant to both the focus and directrix.

Understanding the directrix can greatly aid you in graphing a precise parabola. It aligns with the vertex and the focus to give the parabola its balanced shape.