Problem 29 - Algebra and Trigonometry

Question

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus: \(F(0,2)\)

Step-by-Step Solution

Answer

The equation of the parabola is \( y = \frac{1}{8}x^2 \).

Step 1: Recall the Parabola Definition

A parabola is a set of all points that are equidistant from a point called the focus and a line called the directrix. Here, the vertex of the parabola is at the origin, (0,0).

Step 2: Determine the Directrix

For a parabola with a vertical axis and vertex at the origin, if the focus is at (0,p), the directrix is y = -p. In this problem, the focus is at (0,2), so p = 2, and the directrix is y = -2.

Step 3: Write the Equation of the Parabola

Using the standard form of a parabola equation with a vertex at the origin and focus at (0,p), we have:\[ y = \frac{1}{4p}x^2 \]Since p = 2:\[ y = \frac{1}{8}x^2 \]

Step 4: Verify the Focus

For the parabola \( y = \frac{1}{8}x^2 \), the focus should be at \( (0,2) \). When the standard form \( y = \frac{1}{4p}x^2 \) gives \( 4p = 8 \), we find p = 2, confirming that our focus is correct.

Key Concepts

Vertex at the Origin

When we talk about having a vertex at the origin, we're discussing a special case in the study of parabolas. The vertex is a point where the parabola changes its direction. In simpler terms, it's the very tip of the curve. Usually, a vertex can be anywhere on the coordinate plane. However, a vertex located precisely at (0, 0) indicates it is centered at the very intersection of the x-axis and y-axis.

This positioning simplifies equations and calculations since we don't need to account for shifting the parabola left, right, up, or down from any other place on the grid. When a parabola's vertex is at the origin, everything references this central point making it neat and tidy for solving equations and understanding the shape's behavior.

Focus and Directrix of a Parabola

The focus and directrix are like the guiding stars for a parabola. Think of the focus as a fixed point inside the curve, and the directrix as a line outside of it. The fascinating property of a parabola is that every point on the curve equidistant to both the focus and the directrix. This unique balance defines the curve's shape.

For a parabola with a vertical orientation and a focus located at (0, p), the directrix will be positioned at y = -p. In this setup:

• The focus provides a point to "pull" the curve towards.
• The directrix acts like a "boundary" ensuring the curve doesn't stray too far.

In our specific example, where the focus is at (0, 2), the directrix equation becomes y = -2, ensuring that the parabola's points line up perfectly between these two reference points.

Standard Form of a Parabola Equation

The standard form of a parabola equation simplifies the way we represent this elegant curve. When a parabola has its vertex at the origin, the equation can be simplified significantly. With a vertical axis of symmetry, the formula becomes:

• For vertical opening parabolas: \( y = \frac{1}{4p}x^2 \)

In this equation, "p" represents the distance from the vertex to the focus (or the directrix in the opposite direction).

Plugging in the value of "p" helps us understand the parabola's width and height more clearly. Specifically, in our exercise with focus at (0, 2), we found that p = 2:

• This gives: \( y = \frac{1}{8}x^2 \)
• The coefficient of \( \frac{1}{8} \) indicates the curve's "openness" or how steeply it climbs.

Understanding the standard form equation allows us to write and interpret many parabolas based on their vertex and focus points.