Problem 5 - College Algebra

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

As the parabola becomes wider, the distance between the focus and directrix increases.

Step 1: Understanding the Parabola Structure

In a parabola, the vertex is equidistant from the focus and the directrix. This distance is denoted as "p". As the parabola becomes wider, the distance between its points increases but the definition of focus and directrix in relation to the vertex remains constant.

Step 2: Relationship Between Focus, Directrix, and Vertex

The functional property of a parabola is defined mathematically by the equation: \[ (x - h)^2 = 4p(y - k) \] where \((h, k)\) is the vertex of the parabola, and "p" represents the distance between the vertex and the focus (and the vertex and the directrix). This means as the parabola becomes wider, implying a larger "p", the focus and directrix adjust positions corresponding to the vertex.

Step 3: Effect of a Wider Parabola

When the parabola graph becomes wider, the value of "p" in the equation \((x - h)^2 = 4p(y - k)\) increases. This means that both the focus and directrix are moving further away from the vertex, maintaining the same distance "p" but in opposite directions to accommodate the change in width.

Key Concepts

Focus and Directrix

In the world of parabolas, two crucial elements are the focus and the directrix. Think of the focus as a special point inside the parabola, while the directrix is a fixed straight line outside. These two are like partners in defining the shape and direction of the parabola.
Here’s how they work together: every point on the parabola is equidistant from the focus and the directrix. This means if you pick any point on the curve, the distance to the focus and to the directrix will be the same. This property is the heart of what makes a parabola unique.

• The focus is positioned inside the curve.
• The directrix is outside the curve, parallel to its opening.
• Both help in determining the parabola's width and sharpness.

As a parabola gets wider, its focus and directrix do too. However, they maintain a constant distance "p" from the vertex, ensuring its perfect parabolic harmony.

Vertex of a Parabola

The vertex of a parabola serves as the pivotal point around which the parabola is shaped. Imagine it like the tip or the lowest point (for a downward-opening parabola) from which the parabola spreads out. The vertex is denoted by the coordinates \((h, k)\).Understanding the role of the vertex is essential:

• It lies exactly midway between the focus and the directrix.
• The vertex is the point where the parabola changes direction.
• It is central to calculating the symmetry and shape of the parabola.

The position of the vertex can significantly change the appearance of the parabola. You can think of it as the anchor or balance point that perfectly aligns the parabola to its focus and directrix.

Parabola Equation

The equation of a parabola is a powerful tool that encapsulates its entire shape. The standard form of a parabola that opens vertically is:\[(x - h)^2 = 4p(y - k)\]Here’s what each term means:

• \((h, k)\) are the coordinates of the vertex.
• "p" is the distance from the vertex to the focus (or vertex to directrix), which determines how "stretched" the parabola is.
• The direction of opening (up or down) depends on the sign in front of the "4p" term.

As the parabola widens, which implies an increase in "p", the focus and directrix move apart. This equation elegantly describes these changes, showing how the geometric properties correlate with the algebraic representation. By adjusting parameters in this equation, you effectively dictate how the parabola will open and extend.