Problem 2 - College Algebra

Question

If the equation of a parabola is written in standard form and \(p\) is positive and the directrix is a vertical line, then what can we conclude about its graph?

Step-by-Step Solution

Answer

The graph of the parabola opens upwards.

Step 1: Recall the Standard Form of a Parabola

A parabola with a vertical directrix is typically represented by the equation \((x-h)^2 = 4p(y-k)\) where \((h,k)\) is the vertex of the parabola and \(p\) is the distance from the vertex to the focus, which is also distance to the directrix.

Step 2: Analyze Positive Parameter 'p'

The value of \(p\) indicates the direction in which the parabola opens. Since \(p > 0\), the parabola will open upwards, away from the directrix, because it is positioned at a positive distance from the vertex.

Step 3: Determine Orientation of the Graph

Since we have determined that \(p > 0\) and the directrix is a vertical line, the axis of symmetry of the parabola is vertical. Thus, the parabola opens upwards, with its vertex below its focus.

Key Concepts

Vertex

The vertex of a parabola is a crucial point as it defines the location from which the parabola originates. Imagine this as the starting position of a parabola that opens up or down.
The vertex is denoted by the coordinates \((h,k)\) when the parabola equation is in the form \((x-h)^2 = 4p(y-k)\). This means that the vertex acts as the center point of the parabola, horizontally at \(x = h\) and vertically at \(y = k\).
Knowing the vertex is essential because:

• It helps in determining the axis of symmetry of the parabola.
• It gives a starting reference point to calculate the position of the focus and the directrix.
• It provides insight into the direction in which the parabola opens.

The vertex also shows whether the parabola is centered or has been shifted from the origin. By examining the values of \(h\) and \(k\), one can understand how far the vertex is from the original center \((0,0)\).

Focus

The focus of a parabola is a special point that affects the shape and position of the parabola.
A simple way to comprehend this is by thinking of the focus as a light source—places on the parabola such as fixtures direct their paths by heading towards the focus.
In the standard form equation \((x-h)^2 = 4p(y-k)\):

• The focus lies on the vertical line passing through \(x = h\).
• It is located at the point \((h, k + p)\) when \(p\) is positive, meaning the parabola opens upwards.

Knowing the focus assists in understanding how open or tight the parabola will be. The larger the distance \(p\), the wider the opening of the parabola. In other words, the focus helps to determine how narrow or wide the parabolic curve spreads. Additionally, the focus explains the parabola's reflective properties, where rays parallel to the axis of symmetry are reflected to pass through the focus.

Directrix

The directrix of a parabola might sound complex, but it's quite straightforward. It's an imaginary line, guide, or a boundary that dictates the balance of distances—the directrix ensures a mirrored arrangement between the other mathematical elements of the parabola.
For the equation \((x-h)^2 = 4p(y-k)\):

• If \(p > 0\), the parabola opens upward and the directrix is defined by the line \(y = k - p\). This line lies beneath the vertex.
• The directrix is always perpendicular to the axis that connects the vertex to the focus.

The task of the directrix is to ensure that every point on the parabola is equidistant from the directrix and the focus. This principle aids in crafting the symmetry of the parabola, confirming that each point maintains this balanced distance property.