Problem 5 - College Algebra with Corequisite Support

Question

As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?

Step-by-Step Solution

Answer

The distance between the focus and directrix increases.

Step 1: Understanding the Problem

The problem asks what happens to the distance between the focus and directrix as a parabola becomes wider. To solve this, we need to understand how a parabola is defined and the relationship between its focus, directrix, and width (known as the 'latus rectum').

Step 2: Defining the Parabola

A parabola is defined by the equation \( y = ax^2 + bx + c \). The standard form \( y = ax^2 \) is derived from this and the parameter \( a \) determines the width of the parabola. If \( a \) is small, the parabola is wider; if \( a \) is large, it is narrower.

Step 3: Relating Focus and Directrix Distance

The distance from the vertex to the focus (and similarly to the directrix) is denoted by \( \frac{1}{4|a|} \). This distance is inversely proportional to \( |a| \), which indicates that as \( |a| \) gets smaller (i.e., the parabola gets wider), this distance increases.

Step 4: Analyzing the Relationship

Therefore, as the parabola becomes wider (\( a \to 0 \)), the value of \( \frac{1}{4|a|} \) increases, which means that both the focus and directrix move further apart from the vertex, increasing their distance.

Key Concepts

Focus and Directrix

A parabola is a unique curve in mathematics that has some special properties and components. Two key elements of a parabola are its focus and directrix. The focus of a parabola is a fixed point located inside the curve, around which the shape of the parabola is formed. On the other hand, the directrix is a line that lies perpendicular to the axis of symmetry of the parabola.

Every point on a parabola is equidistant from the focus and directrix. This relationship is fundamental to the definition of a parabola. As such, understanding the focus and directrix can help us analyze various other aspects of a parabola, including its width and shape. The distance between the focus and the directrix plays a crucial role in the geometric properties of a parabola, especially when examining how the parabola expands or contracts.

Latus Rectum

The latus rectum of a parabola is a line segment that passes through the focus and is perpendicular to the axis of symmetry. One of the distinct features of the latus rectum is that its length gives a sense of how wide the opening of the parabola is at the focus.

The formula for the length of the latus rectum is given by the fraction \( \frac{1}{|a|} \), where \( a \) is the coefficient of the \( x^2 \) term in the equation of the parabola \( y = ax^2 + bx + c \). Therefore, the length of the latus rectum is inversely related to the absolute value of \( a \).

• If \( a \) is small, meaning the parabola is broad, the latus rectum is long.
• If \( a \) is large, meaning the parabola is narrow, the latus rectum is short.

The length of the latus rectum essentially measures how wide the parabola is at the focus, making it a vital component in parabola geometry.

Vertex of a Parabola

The vertex of a parabola serves as the point where the curve changes direction and is an essential part of understanding parabola geometry. In the standard form of a parabolic equation, \( y = ax^2 + bx + c \), the vertex can be found using the formula:

• \( x = -\frac{b}{2a} \)
• Substituting back, \( y = a(-\frac{b}{2a})^2 + b(-\frac{b}{2a}) + c \)

The vertex not only represents the lowest or highest point on the graph of the parabola, depending on the sign of \( a \), but it also acts as the midpoint between the focus and the directrix.

This characteristic makes the vertex an important element when examining the properties of a parabola, as distances from the vertex to the focus and the directrix are key in calculating the defining dimensions of the parabola.

Width of a Parabola

The width of a parabola is fundamentally determined by its coefficient \( a \) in the equation \( y = ax^2 + bx + c \). When we talk about widening or narrowing a parabola, we are referring to how far apart or close the arms of the parabola are.

An important factor to remember is:

• If \( |a| \) is small, the parabola appears wider or broader.
• If \( |a| \) is large, the parabola appears narrower.

The width of the parabola affects its shape, and this is crucial when evaluating the spatial aspect of the parabola, as well as its interaction with the focus and directrix. Therefore, understanding how \( |a| \) influences the width provides insights into the geometric relationship between the focus, directrix, and overall appearance of the parabola.