Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(y=4\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(x^2 = 16y\).

Step 1: Understand the characteristics of a parabola

For any parabola, its standard equation comes in two forms: 1. \(y^2 = 4px\) if the parabola opens towards either the positive x-axis or negative x-axis2. \(x^2 = 4py\) if the parabola opens towards either the positive y-axis or negative y-axis. The value of 'p' denotes the distance from the vertex to the focus or from the vertex to the directrix. If the parabola opens upwards or to the right, 'p' is positive. If it opens downwards or to the left, 'p' is negative.

Step 2: Identify the direction in which the parabola opens

Since the directrix is a horizontal line, the parabola opens in a vertical direction. The line \(y=4\) lies above the origin. Because every point on the parabola is closer to the focus than to the directrix, the parabola opens in the opposite direction, i.e., downwards. Hence, the parabola will have a negative 'p' value and its equation is in the form \(x^2 = -4py\).

Step 3: Identify the value of 'p'

The value of 'p' in the standard equation of the parabola is the distance from the vertex to the focus or the vertex to the directrix. Its absolute value is the distance from the vertex to the line \(y = 4\). The vertex of the parabola is (0,0) and the directrix is 4 units away from y. So, the absolute value of 'p' is 4. As the parabola opens downwards, 'p' is negative. Therefore, 'p' = -4.

Step 4: Write the standard form of the equation of the parabola

Substitute 'p' into the standard form of the equation for the parabola. The equation of the parabola is \(x^2 = -4(-4)y\), which simplifies to \(x^2 = 16y\).

Key Concepts

Standard Form

The standard form of a parabola's equation is crucial for understanding its shape and orientation. For parabolas, the standard form can be either\( x^2 = 4py \) or\( y^2 = 4px \). These forms help in identifying which direction the parabola opens.
When the equation is in the form \( x^2 = 4py \), it means the parabola opens vertically, either up or down. Conversely, \( y^2 = 4px \) indicates a horizontal opening, either left or right. The variable 'p' in these equations represents the distance between the vertex and the directrix or the focus. It's vital in determining the parabola's orientation—whether it opens upwards, downwards, left, or right—depending if 'p' is positive or negative.
Understanding the standard form allows you to quickly assess these characteristics and better visualize the parabola.

Vertex

The vertex of a parabola is the pivotal point where it changes direction, making it essential for graphing or analyzing a parabola. It is typically represented as a singular point \((h, k)\) in the coordinate plane. For a parabola defined by the equation\( x^2 = 4py \), the vertex is usually at the origin, shown as \((0, 0)\).
In more complex functions, you might see a shifting of the vertex defined as \((h, k)\) by transforming the equation to \((x-h)^2 = 4p(y-k)\). Here, you move the entire parabola horizontally to 'h' and vertically to 'k'.
Understanding the vertex is crucial for plotting parabolas and for determining other properties like the axis of symmetry and the opening direction.

Directrix

The directrix of a parabola is an imaginary line used to define its shape and position. It is a fixed, straight line that, together with the focus, guides the creation of the parabola. Each point on the parabola is equidistant from the focus and this line.
For a vertically opening parabola, the directrix will have a horizontal equation like\( y = k - p \) for upward openings, and \( y = k + p \) for downward. The location of the directrix relative to the vertex specifies the parabola's width and flex.
When solving the exercise of finding the parabola given a directrix\( y = 4 \) and a vertex at the origin\((0, 0)\), the directrix helps determine the shift and distance (or 'p') of 4 units downward, finding 'p' crucial for constructing the standard equation.

Vertical Parabola

A vertical parabola is one whose graph opens either upwards or downwards. This depends largely on the equation structure \( x^2 = 4py \) where it signifies a vertical orientation.

• If the value of 'p' is positive, the parabola opens upward.
• If 'p' is negative, it opens downward.

The directrix influences the direction of 'p'. For example, in our problem with a given directrix of \( y = 4 \), the vertex at \((0,0)\), and the parabola opening downward, results in \( p = -4 \).
Vertical parabolas are commonly seen in problems of projectile motion or satellite dishes. Their symmetry about the y-axis is another identifying characteristic, crucially impacting everything from their appearance to their functional applications. Understanding this can help identify properties and behaviors uniquely attributed to vertically oriented parabolas.