Question

For the following exercises, sketch the graph of each conic. $$ x^{2}=12 y $$

Short answer

The parabola \( x^2 = 12y \) opens upward with vertex at \((0,0)\), focus at \((0,3)\), and directrix \( y = -3 \).

Step-by-step solution

Identify the Conic

Given the equation is \( x^2 = 12y \), notice that it resembles the standard form of a parabola, \( x^2 = 4py \). Here, the squared term is \( x^2 \), indicating a vertical parabola (opening up or down).

Determine the Parameters

The equation \( x^2 = 12y \) can be rewritten as \( x^2 = 4(3)y \). Comparing this with the standard form \( x^2 = 4py \), we find \( 4p = 12 \), thus \( p = 3 \). This tells us that the distance from the vertex to the focus is 3 units.

Identify the Vertex

For the parabola equation \( x^2 = 4py \), the vertex is at the origin \((0, 0)\) since there are no terms shifting it horizontally or vertically from the standard position.

Locate the Focus and Directrix

Since \( p = 3 \), the focus of the parabola is located at \((0, 3)\) since the parabola opens upwards. The directrix, being \( p \) units below the vertex, is the line \( y = -3 \).

Sketch the Parabola

Plot the vertex at the origin \((0,0)\). Mark the focus at \((0,3)\) and draw the directrix as the horizontal line \( y = -3 \). Since the parabola opens upward, sketch a symmetric curve extending upwards from the vertex.

Key concepts

Parabola

A parabola is a specific type of conic section—derived from slicing a cone in a particular way. It is defined mathematically as the set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix.

Parabolas have a characteristic U-shape, which can open upwards, downwards, or sideways, depending on the orientation of the squared term. In the case of the equation \(x^2 = 12y\), it is a vertical parabola that opens upward because the squared term is \(x^2\).

The general form of a vertical parabola is \(x^2 = 4py\). Here, \(p\) represents the distance from the vertex to the focus, giving us a clear indication of how wide or narrow the parabola appears. Understanding these elements is crucial when graphing parabolas or solving related mathematical problems.

Graph Sketching

Graph sketching involves drawing a rough representation of a graph based on the key features of a mathematical equation or function. For parabolas, like in the equation \(x^2 = 12y\), key features include the vertex, the axis of symmetry, and the direction in which the parabola opens.

To start sketching a parabola:

• Identify the vertex. For a standard parabola \(x^2 = 4py\), the vertex is located at the origin \( (0, 0) \), unless the graph is shifted.
• Mark the focus and the directrix, which guide you in drawing the parabola's precise shape.
• Draw the axis of symmetry, which is a vertical line through the vertex.
• The parabola will always be symmetric about this axis.

By noting symmetry and key points, such as the vertex and focus, you can sketch an accurate graph that represents the behavior of the parabola.

Focus and Directrix

The focus and directrix are two fundamental components that define the parabola. The focus is a fixed point located at a certain distance from the vertex behind the curve, while the directrix is a fixed line situated on the opposite side of the vertex, the same distance away.

In the equation \(x^2 = 12y\), as it matches the form \(x^2 = 4py\), we find \(p = 3\). This means:

• Focus is located at \( (0, 3) \), 3 units above the vertex.
• Directrix is the line \( y = -3 \), 3 units below the vertex.

The focus concentrates all points of the parabola such that each point on the parabola is equidistant from both the focus and the directrix. This property allows the curve to maintain its shape consistently along the length of the parabolic path.

Vertex of a Parabola

The vertex is essentially the peak point or the "tip" of the parabola, serving either as its maximum or minimum point, depending on the orientation.

In a vertically oriented parabola, like in \(x^2 = 12y\), the vertex provides the point about which the parabola's graph is symmetric. Here, it is located at \( (0, 0) \).

The vertex's role is crucial as it often forms the starting point in graph sketching. From the vertex, distances are measured to determine the positions of the focus and directrix. These distances, expressed by \(p\), help shape the graph's appearance by defining the spread and width of the parabola, ensuring that all points on the curve are symmetric around the vertex.