Question

For the following exercises, sketch the graph of each conic. $$ y^{2}=20 x $$

Short answer

Sketch a right-opening parabola with vertex (0,0), focus (5,0), and directrix at x=-5.

Step-by-step solution

Identify the Type of Conic

The given equation is \( y^2 = 20x \). This equation is in the form \( y^2 = 4px \), which represents a parabola that opens to the right. This is because \( y \) is squared, and \( x \) is linear.

Determine the Vertex of the Parabola

In the equation \( y^2 = 4px \), the vertex is at the origin \((0,0)\) for the given equation \( y^2 = 20x \). Therefore, the vertex of this parabola is \((0, 0)\).

Calculate the Parameter for the Focal Length

Using the equation \( y^2 = 4px \), we find \( 4p = 20 \). Solving for \( p \), we get \( p = 5 \). This means the focus of the parabola is 5 units to the right of the vertex at \((5,0)\).

Determine the Directrix

The directrix of the parabola is a vertical line \( x = -5 \), located 5 units to the left of the vertex, opposite the focus.

Sketch the Graph

The parabola opens to the right with the vertex at \((0,0)\), focus at \((5,0)\), and directrix at \( x = -5 \). Draw the vertex and plot the focus and directrix on the graph. Sketch the parabola by drawing a symmetric curve opening to the right around the axis through the focus.

Key concepts

Parabola

A parabola is a unique type of conic section that resembles a "U" or an inverted "U," which can be either upwards, downwards, or sideways. Conic sections are curves obtained by cutting a cone with a plane. A parabola is formed when a plane is parallel to the surface of the cone. In the equation \( y^2 = 20x \), we have a sideways opening parabola. Here, the parabolic shape is determined by whether the \( x \) or \( y \) term is squared. If \( y \) is squared and \( x \) is linear, the parabola opens along the \( x \)-axis. Conversely, if \( x \) is squared, it will open along the \( y \)-axis. Parabolas have some essential characteristics and parts such as a vertex, a focus, and a directrix.

Vertex

The vertex is an important feature of a parabola, serving as its peak or "tip." It is precisely where the curve changes direction. In mathematical terms, the vertex is the point where the conic section intersects its axis of symmetry. In the equation \( y^2 = 20x \), the vertex is located at the origin, \((0, 0)\). This particular point is significant because it represents the position from which every other part of the parabola is symmetrically balanced. Due to its position at the origin, the parabola in the problem is symmetric about the horizontal axis through this vertex.

Focus

The focus of a parabola is a special point from which distances to any point on the curve are measured. It is located inside the parabola and acts as an internal guidepost for the curve's shape. This aspect of a parabola is critical because any line drawn perpendicular to it and extending to the curve reflects the same length on either side—that is a property of a parabola. In our given equation \( y^2 = 4px \) where \( 4p = 20 \), we find \( p = 5 \), which places the focus at \((5,0)\). Five units to the right of the vertex, it defines how "wide" or "narrow" the parabola is perceived by the viewer.

Directrix

The directrix is part of the geometric description of a parabola and is crucial for understanding its symmetrical nature. It is a line located opposite the focus and is used as a reference line. For our parabola described by the equation \( y^2 = 20x \), the directrix is the vertical line \( x = -5 \). This line is situated 5 units to the left of the vertex, mirroring the distance of the focus to the right. The directrix is equidistant from the vertex, just like the focus, which is a unique property that helps maintain the shape of the parabola and define its symmetry.