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

Step 1: Compare to standard form

The given equation is \(x^2 = 12y\). The standard form of a vertically oriented parabola is \(x^2 = 4py\). By comparing the given equation to the standard form, we identify that \(4p = 12\).

Step 2: Solve for p

To solve for \(p\), divide both sides of the equation \(4p = 12\) by 4. Thus, \(p = 3\).

Step 3: Determine the focus

For a parabola \(x^2 = 4py\), the vertex is at the origin \((0,0)\), and the focus is at \((0, p)\). With \(p = 3\), the focus is at \(F(0, 3)\).

Step 4: Determine the directrix

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

Key Concepts

Parabola

A parabola is a unique locus of points in a plane, characterized by its symmetrical open curve. It has distinct properties that differentiate it from other conic sections like circles and ellipses.
Parabolas can open upwards, downwards, or sideways depending on their orientation.
If the parabola is positioned vertically, the opening could either be upward or downward, based on its sign. If it’s positive, it opens upwards, if negative, it goes downwards. For horizontally oriented parabolas, depending on the sign, they either open rightwards or leftwards.

• The symmetry is significant in parabolas, meaning both halves mirror each other.
• The vertex is an important point located at the base of this symmetry line.

Understanding these characteristics helps in graphing and solving equations involving parabolas.

Focus and Directrix

Understanding the focus and directrix of a parabola is essential in comprehending its geometric structure.
The focus is a fixed point inside the parabola where all points on the curve maintain a constant distance that is proportional to the directrix.
The directrix, on the other hand, is a line that lies outside of the parabola and is perpendicular to the axis of symmetry. This line assists in defining the unique shape of the parabola.

• The focus is located inside the parabola, on the axis of symmetry.
• The directrix runs parallel to the direction in which the parabola opens.
• Understanding that the distance from any point on the parabola to the focus equals the perpendicular distance from that point to the directrix is key in its definition.

In the problem example, with the equation \(x^2 = 12y\), the vertex is at the origin \((0,0)\), the focus is at \(F(0,3)\) and the directrix at \(y=-3\). This means the parabola is vertically oriented and opens upwards.

Standard Form of Parabola

The standard form for the equation of a parabola offers a simplified representation of its curve, focusing on its orientation and position in the coordinate plane.
For parabolas that open vertically, the standard form is \(x^2 = 4py\). In this format, \(p\) is a constant that determines the distance between the vertex and the focus or directrix, indicating the parabola's width and direction.

• Here, \(x^2\) suggests the parabola’s vertical orientation.
• The constant \(4p\) helps to find the appropriate values for the focus and directrix.
• It reveals that the vertex is directly at the origin, simplifying graph representation.

The conversion from general forms to standard forms is crucial. By focusing on \(x^2 = 12y\), it aligns directly with \(x^2 = 4py\) upon understanding that \(4p = 12\). Solving yields \(p = 3\), providing essential details for graphing and analysis, such as the focus being \(F(0, 3)\), and the directrix \(y = -3\). This standard form makes analysis simpler and results more digestible for further applications.