Problem 29 - Precalculus Mathematics for Calculus

Question

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 \(x^2 = 8y\).

Step 1: Understand the Given Information

We have a parabola with a vertex at the origin, \((0,0)\). We are also given the focus of the parabola, which is \(F(0, 2)\). The typical form of a parabola with a vertical axis is \(x^2 = 4py\), with \(p\) being the distance from the vertex to the focus.

Step 2: Identify the Parameter p

Since the focus is \((0, 2)\) and the vertex is at the origin \((0,0)\), the distance \(p\) from the vertex to the focus is 2. Thus, \(p = 2\).

Step 3: Write the Equation of the Parabola

The general formula for a parabola with a vertical axis centered at the origin is \(x^2 = 4py\). By substituting \(p = 2\) into this formula, we get \(x^2 = 4 \times 2 \times y\), which simplifies to \(x^2 = 8y\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point, as it represents the point where the parabola changes direction. When a parabola opens upwards or downwards, the vertex will be the lowest or highest point, respectively. In this kind of parabola, the vertex is typically represented as the point

• the equation can be simplified to
• turns the parabola either upwards or downwards

For the parabola in our exercise, the vertex is at the origin which means the parabola peaks at the starting point, or zero point, in both the x and y directions.

Focus of a Parabola

The focus of a parabola is a fixed point that, combined with a directrix, helps define the curve. It lies along the axis of symmetry of the parabola. When a parabola is plotted on a coordinate plane, any point

• is approximately equidistant from the focus
• and the directrix, a line that does not intersect the parabola itself.

In our particular example, we know that the focus is positioned at which implies that it is positioned directly above the origin. The focus, therefore, plays a critical role in determining the shape and equation of the parabola.

Distance from Vertex to Focus

The distance from the vertex to the focus of a parabola, often denoted as "", determines several aspects about the parabola including its "width" or "narrowness." The greater the distance

• the broader the parabola
• is said to open with a larger spread.

In the given exercise, the distance, , is determined by the focus point When the vertex is at the origin such as here, it becomes relatively simple to determine the value of which directly influences the derived equation Here, meaning the parabola expands as it propagates vertically through the coordinate plane.