Problem 25 - Precalculus with Calculus Previews

Question

In Problems \(25-44,\) find an equation of parabola that satisfies the given conditions. Focus \((0,7),\) directrix \(y=-7\)

Step-by-Step Solution

Answer

The parabola equation is \(x^2 = 28y\).

Step 1: Understanding the Properties of Parabola

A parabola is defined by its focus and directrix. The focus is a point represented as \((0,7)\), and the directrix is a line given by \(y = -7\). The vertex of the parabola lies midway between the focus and the directrix.

Step 2: Determine the Vertex

The vertex is midway between the focus at \((0,7)\) and the directrix \(y = -7\). The midpoint is the average of the y-coordinates of the focus and directrix: \[\frac{7 + (-7)}{2} = 0\]. Thus, the vertex of the parabola is at \((0,0)\).

Step 3: Find the Distance from Vertex to Focus/Directrix

The distance from the vertex \((0,0)\) to the focus \((0,7)\) or the directrix \(y = -7\) is 7 units. This distance is represented as \(p\), which in this case, \(p = 7\).

Step 4: Write the Standard Form Equation

Since the parabola opens vertically and the vertex is \((0,0)\), the equation has the form \(x^2 = 4py\). Substitute \(p = 7\) to get\[x^2 = 4 \cdot 7 \cdot y\]or \(x^2 = 28y\). Thus, the equation of the parabola is \(x^2 = 28y\).

Key Concepts

Focus and Directrix

The focus and directrix are key elements that define the structure of a parabola. The focus is a specific point, in this case, (0,7), where the parabola curves around. The directrix is a straight line, represented by the equation \(y = -7\), that helps in locating the position and orientation of the parabola. Together, the focus and directrix arrange so that any point on the parabola is equidistant to both the focus and this line. This unique relationship defines the parabola's perfect symmetrical shape.

• Focus: The feature point on the interior of the parabolic curve.
• Directrix: A linear boundary placed outside the curve that forms a perpendicular alignment with the parabola's principal axis.

The distance from the vertex to the directrix (or to the focus) is aligned perpendicularly, ensuring symmetry on both sides of the axis. Understanding the relationship between these two components is essential when identifying and constructing parabolas.

Vertex of Parabola

The vertex of a parabola is the center point at which the curve makes its sharpest turn. For the parabola described in this exercise, the vertex is at (0,0). This point is exactly midway between the focus (0,7) and the directrix line \(y = -7\).

• The vertex is always located on the axis of symmetry.
• In a perfectly vertical or horizontal parabola, the vertex serves as the graph's "start point" from which it opens.

For those learning about parabolas, grasping the vertex concept helps in visualizing how the curve is formed and ensures accurate plotting on a graph. It also provides the initial step towards deriving equations, as it transparently links the focus and directrix.

Standard Form Equation

The standard form equation for a vertically oriented parabola is given as \( x^2 = 4py \). This form reveals essential characteristics like the parabola's direction and its stretching factor. In this exercise, the parabola has a vertex at the origin (0,0), simplifying the formulation of its equation.

• Substituting \(p\): Here, \(p\) is the distance from the vertex to the focus, measured as 7 units.
• Resulting Equation: By placing \(p = 7\) into the equation, we derive \( x^2 = 28y \).

This indicates that the parabola opens upwards, with its axis aligned vertically. Understanding the standard equation form empowers students to transform geometric relationships between the focus, directrix, and vertex into algebraic expressions that define the parabola.