Problem 2 - Algebra and Trigonometry

Question

The graph of the equation \(x^{2}=4 p y\) is a parabola with focus F(_____,_____) and directrix \(y=\) _____. So the graph of \(x^{2}=12 y\) is a parabola with focus \(F\) (_____,_____) and directrix \(y=\) _____.

Step-by-Step Solution

Answer

Focus: \((0,3)\), Directrix: \(y=-3\).

Step 1: Understanding the Standard Equation

The standard form of a parabola that opens upwards is \(x^2 = 4py\), where \(p\) is the distance from the vertex to the focus and also from the vertex to the directrix.

Step 2: Identify the Value of p

In the equation \(x^2 = 12y\), we need to rewrite it in the form \(x^2 = 4py\). Comparing, we have \(4p = 12\).

Step 3: Solve for p

Divide both sides by 4: \(p = \frac{12}{4} = 3\). So, \(p = 3\). This means the distance from the vertex to the focus and the vertex to the directrix is 3 units.

Step 4: Find the Focus

Since the parabola opens upwards, the vertex is at the origin (0, 0). The focus is \(p\) units above the vertex. So, the focus \(F\) is at \((0, 3)\).

Step 5: Find the Directrix

The directrix is a horizontal line \(p\) units below the vertex of the parabola. Since \(p = 3\), the directrix is \(y = -3\).

Key Concepts

Focus of a Parabola

In a parabola, the focus is a specific point internally positioned along the axis of symmetry. For problems involving the equation in the form \(x^2 = 4py\), the focus dictates how the parabola bends and opens.
The parabola behaves in such a way that each point on it is equidistant from the focus and the directrix.

• For the given equation \(x^2 = 12y\), we've identified the value \(p = 3\).
• This means the focus is 3 units above the vertex in the vertical direction.
• Thus, for this parabola, the focus is positioned at \((0, 3)\).

Understanding the focus helps grasp strong spatial relationships within the graph.

Directrix of a Parabola

The directrix of a parabola is a straight line that works alongside the focus to define its shape. It is perpendicular to the axis of symmetry and is located \(p\) units away from the vertex on the opposite side of the focus.
For the parabola represented by the equation \(x^2 = 4py\):

• The directrix is aligned horizontally if the parabola opens upwards or downwards.
• For our parabola \(x^2 = 12y\) with \(p = 3\), the directrix is positioned 3 units below the vertex.
• This provides the equation for the directrix as \(y = -3\).

The directrix plays a crucial role in maintaining the symmetry and the reflective characteristic of a parabola.

Vertex of a Parabola

In any parabola, the vertex is the central point known as the basis for its symmetry. It marks the intersection of the parabola's axis of symmetry, and often the highest or lowest point depending on its orientation (opening up or down).
For our given parabola \(x^2 = 12y\):

• The vertex is naturally at the coordinate \((0, 0)\) when expressed in standard form.
• Since the parabola opens upwards in this case, the vertex is at the lowest point.

Knowing the vertex is fundamental for graphing, allowing us to locate the core point from where the parabola grows.

Equation of a Parabola

The equation of a parabola takes the form \(x^2 = 4py\) for those opening vertically. This equation relates directly to the parabolic features like the focus and directrix through the constant \(p\).
In our scenario with \(x^2 = 12y\):

• You begin by comparing it with the general form \(x^2 = 4py\).
• By setting \(4p = 12\), we find that \(p = 3\).
• This simple form enhances understanding, as \(p\) gives the distance to the focus and directrix.

Understanding this setup allows one to derive significant properties and confirms the structural aspects of the parabola.