Question

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$ y^{2}=4 x $$

Short answer

Focus: (1,0). Directrix: x = -1.

Step-by-step solution

Identify the form of the equation

The given equation is \( y^2 = 4x \), which is a parabola that opens horizontally. This matches the standard form \( y^2 = 4px \), where \( p \) determines the distance from the vertex to the focus and the directrix.

Determine the value of p

By comparing \( y^2 = 4x \) to the standard form \( y^2 = 4px \), we see that \( 4p = 4 \). Solving for \( p \), we get \( p = 1 \).

Find the vertex of the parabola

For \( y^2 = 4px \), the vertex is at the origin \((0,0)\).

Locate the focus

Since the parabola opens to the right and \( p = 1 \), the focus is located \( p \) units to the right of the vertex. Thus, the focus is at \( (1,0) \).

Write the equation of the directrix

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

Make a sketch

Draw a horizontal parabola opening to the right with the vertex at \((0,0)\). Plot the focus at \((1,0)\) and the directrix line at \( x = -1 \).

Key concepts

Focus of a Parabola

The focus of a parabola is a specific point located in relation to the vertex. It plays an integral role in defining the shape and direction of the parabola. In the equation of a parabola that opens horizontally, like in the form \( y^2 = 4px \), the focus is found \( p \) distance from the vertex. Here, \( p \) represents the distance from the vertex to the focus along the axis of symmetry of the parabola.
For example, in the parabola described by the equation \( y^2 = 4x \):

• We find \( p = 1 \) by comparing it to the standard form \( y^2 = 4px \).
• The focus is then located at \((p, 0) = (1, 0)\) because the parabola opens to the right.

Therefore, understanding the location of the focus helps in sketching and analyzing the parabola's structure.

Directrix of a Parabola

The directrix of a parabola is a crucial component that, together with the focus, defines the parabola's properties. It is a straight line that is perpendicular to the axis of symmetry of the parabola.
For parabolas in the form \( y^2 = 4px \), like our example \( y^2 = 4x \), the directrix is located at a distance \( p \) units away from the vertex, opposite the focus. This means:

• If the parabola opens to the right, as in \( y^2 = 4x \), the directrix will be a vertical line to the left of the vertex.
• So, for \( p = 1 \), the directrix is at \( x = -1 \) since it is \( p \) units left of the vertex \((0, 0)\).

The directrix, together with the focus, ensures that any point on the parabola is equidistant from the focus and the directrix.

Vertex of a Parabola

The vertex of a parabola is the point where the parabola changes direction and is either the highest or lowest point, depending on its orientation. In equations like \( y^2 = 4px \), the vertex often serves as the reference point for locating both the focus and the directrix. For the equation \( y^2 = 4x \), the vertex is straightforward to determine:

• The vertex is at the origin, \((0,0)\), because there is no constant term shifting the parabola from this position.
• This vertex is the midpoint between the focus \((1,0)\) and the directrix \( x=-1 \), serving as the base from which the parabola expands.

Being able to identify the vertex is crucial for graphing the parabola and understanding its orientation in space.

Equation of a Parabola

The equation of a parabola gives us critical information about its shape and direction. Parabolas that open sideways, like in the form \( y^2 = 4px \), have an axis of symmetry that is horizontal. The distance \( p \) determines how wide or narrow the parabola is and its direction from the vertex.
In our example, the equation:

• \( y^2 = 4x \) indicates a horizontal parabola opening to the right.
• The coefficient 4 represents \( 4p \) where \( p \) is the distance from the vertex to the focus and directrix, calculated as \( p = 1 \).

Grasping the equation of a parabola is essential for predicting and understanding how the parabola behaves.