Problem 9 - Algebra 2

Question

Write an equation of a parabola with a vertex at the origin and the given focus. focus at \((0,7)\)

Step-by-Step Solution

Answer

The equation of the parabola with a vertex at the origin and a focus at \((0,7)\) is \(y = (1/28)x^2\).

Step 1: Find the Distance Between the Vertex and the Focus

The vertex is at the origin \((0,0)\), and the focus is at \((0,7)\). These points are on the y-axis, so the distance between them is simply the difference in their y-coordinates, which is \(7-0 = 7\). So, \(p = 7\).

Step 2: Find the Value of \(a\)

With a value for \(p\), we can now solve for \(a\). The value of \(a\) in the formula for a parabola is equal to \(1/(4p)\). So substituting for \(p\), we have \(a = 1/(4*7) = 1/28\).

Step 3: Write the Equation of the Parabola

The general form of the equation for a parabola that opens upwards is \(y = ax^2\). Substituting the value for \(a\) that we found in the previous step, we get \(y = (1/28)x^2\). This is therefore the equation of the parabola that meets the conditions given in the problem.

Key Concepts

Understanding the Vertex of a Parabola

The vertex of a parabola is a crucial concept. It is the point where the parabola changes direction and is often considered its "tip."
In coordinate geometry, the vertex is expressed as a point \( (h, k) \), where \( h \) and \( k \) are the coordinates. For the problem we are solving, the vertex of the parabola is at the origin, \( (0, 0) \). This means the parabola is symmetrically aligned about the origin, with the axis of symmetry being either the x-axis or the y-axis.
Since the vertex is at such a central point, it often simplifies calculations, especially when finding the focus and directrix.

Exploring the Focus of a Parabola

The focus of a parabola is another essential point. It lies along the axis of symmetry of the parabola and determines how "wide" or "narrow" the parabola is. In this problem, the focus is located at \( (0, 7) \).
This tells us that the parabola opens upwards. The distance from the vertex to the focus is denoted as \( p \), which is a parameter in the parabola's equation. Since the focus is \( 7 \) units above the vertex, \( p = 7 \).
The vertex and focus together allow us to compute the value of \( a \), which appears in the equation of the parabola, thereby affecting the shape and directrix of the parabola.

Using the Distance Formula

The distance formula is a valuable tool in coordinate geometry to calculate the distance between two points in a plane. In this exercise, it is used to find the distance between the vertex \( (0, 0) \) and the focus \( (0, 7) \) of the parabola.

• The formula is given by: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Since both points lie on the y-axis, only the difference between the y-coordinates matters.
Using the formula simplifies to \( 7 - 0 = 7 \), which is simply \( 7 \).
This distance is crucial as it directly contributes to determining the parameter \( p \) of the parabola.

Basics of Coordinate Geometry in Parabola

Coordinate geometry, also known as analytic geometry, blends algebra and geometry to describe geometric figures. It uses coordinates to define shapes, equations to describe curves, and calculations to find distances and angles.
For a parabola, it helps us describe the exact location of the vertex and the focus with precision in a two-dimensional plane.
In this exercise, coordinate geometry provides a clear method to determine the orientation and dimensions of the parabola by systematically calculating the distance between the vertex and the focus and subsequently using it to derive the equation of the parabola. Understanding these calculations helps in visualizing and sketching the curve, which is key in many branches of mathematics and applied sciences.