Problem 2 - Precalculus: Mathematics for Calculus

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: Identify the parabola properties

The general equation 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. For this equation, the vertex is at the origin \((0,0)\).

Step 2: Determine the given parameters

You are given \(x^2 = 12y\). In this case, compare it to the standard form \(x^2 = 4py\). Here, \(4p = 12\).

Step 3: Solve for p

Solve \(4p = 12\) to find \(p\). Divide both sides of the equation by 4: \(p = \frac{12}{4} = 3\).

Step 4: Locate the focus

The focus of the parabola \(x^2 = 4py\) is at \((0, p)\). Given that \(p = 3\), the focus of the parabola is at \((0, 3)\).

Step 5: Identify the directrix

The directrix of the parabola \(x^2 = 4py\) is given by the equation \(y = -p\). With \(p = 3\), the directrix is \(y = -3\).

Key Concepts

Focus of a Parabola

In geometry, the focus of a parabola is a significant point that is directly related to how the parabola is shaped. For a parabola that opens upwards, like the one given by the equation \(x^2 = 4py\), the focus lies on the vertical axis (y-axis). The precise location of the focus is determined by the parameter \(p\), which represents the distance from the vertex to the focus.

• The general formula for the focus of a parabola opening upwards is \((0, p)\).
• The vertex of the parabola is at the origin \((0, 0)\).
• In the equation \(x^2 = 12y\), comparing it with the standard form \(x^2 = 4py\), \(p\) is determined to be 3.
• Thus, the focus of this parabola is at the point \((0, 3)\).

Understanding the focus helps in visualizing the parabola, indicating where the parabolic curve would "focus" light or sound if it were a physical object.

Directrix of a Parabola

The directrix is another fundamental element of a parabola. It acts as a reference line that helps define the width and orientation of the parabola. While the focus is within the curve, the directrix is outside the curve, opposite the focus. For the equation \(x^2 = 4py\), this line is horizontal.

• The directrix acts as the spread constraint for the parabola.
• The mathematical formula for the directrix of a parabola that opens upwards is \(y = -p\).
• Given \(p = 3\), as calculated from the equation \(x^2 = 12y\), this directrix becomes \(y = -3\).
• The presence of both focus and directrix allows us to define the parabolic curve completely.

This relationship forms the geometric property that every point on the parabola is equidistant to the focus and the directrix.

Properties of Parabolas

Parabolas have many fascinating properties that make them a widely studied topic in Geometry and Algebra. Not only are they present in mathematical problems, but their applications extend to real-life scenarios, such as satellite dishes and headlights. Understanding the basic properties allows for better comprehension of these applications.

• A parabola is a symmetric curve, and its symmetry axis is the vertical line passing through the vertex.
• The equation of a standard parabola that opens upwards is \(x^2 = 4py\).
• Its geometric structure means that any point on the parabola is the same distance from the focus and the directrix.
• As the value of \(p\) increases, the parabola becomes wider; conversely, a smaller \(p\) results in a more "narrow" shape.

These properties ensure the parabola's unique place in mathematics, making it strong enough to handle reflecting properties, perfect for concave-shaped objects to collect or emit energy dynamically.