Problem 79 - A Graphical Approach to College Algebra

Question

Write an equation for each parabola with vertex at the origin. Through \((2,-2 \sqrt{2}) ;\) opening to the right

Step-by-Step Solution

Answer

The equation of the parabola is \(y^2 = 4x\).

Step 1: Understand the Vertex Form

For a parabola opening to the right with vertex at the origin, the general equation is given by \( y^2 = 4px \), where \( p \) is the distance from the vertex to the focus. Here, we know the vertex is at the origin \((0,0)\).

Step 2: Substitute the Given Point

The parabola passes through the point \((2, -2\sqrt{2})\). Substitute \(x = 2\) and \(y = -2\sqrt{2}\) into the general equation \(y^2 = 4px\).

Step 3: Solve for p

Substitute \(x = 2\) and \(y = -2\sqrt{2}\) into the equation: \((-2\sqrt{2})^2 = 4p(2)\). Simplify the equation to find \(p\).

Step 4: Compute and Simplify

Calculate \((-2\sqrt{2})^2 = 8\), and then set it equal to \(8p\). Thus, we have \(8 = 8p\), which simplifies to \(p = 1\).

Step 5: Write the Equation of the Parabola

With \(p = 1\), substitute back into the general form \(y^2 = 4px\) to get \(y^2 = 4x\).

Key Concepts

Vertex Form

The vertex form of a parabola is a way to express its equation, making it easier to identify key features like the vertex. The general vertex form equation for a parabola can be given as \[ y = a(x-h)^2 + k \] for parabolas that open upwards or downwards, and \[ x = a(y-k)^2 + h \] for those that open sideways. Here,

• \( (h, k) \) is the vertex of the parabola
• \( a \) determines the direction and width of the parabola

In this problem, we consider a parabola opening to the right with the vertex at the origin, so the equation is simplified to \[ y^2 = 4px \] where

• \( p \) is the distance from the vertex to the focus.

For this exercise, understanding how the given point \( (2, -2\sqrt{2}) \) fits into the vertex form allows us to determine the value of \( p \) and ultimately write down the specific equation of the parabola in vertex form. It is versatile because it highlights the vertex, a key part of the parabolic shape.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, uses algebra to study geometric concepts through the use of coordinates on a plane. Parabolas, which are part of the set of "conic sections," can be precisely studied using this method. The power of this approach is that it allows for the analysis and representation of shapes like parabolas using algebraic equations.

In this exercise, understanding coordinate geometry helps in substituting the given point into the parabola's equation to find specific parameters. By substituting points like \( (2, -2\sqrt{2}) \) into the equation \( y^2 = 4px \), we can solve for \( p \). This showcases how coordinate geometry bridges algebra and geometry, allowing us to understand how the points connect to form parabolas. These coordinates map out the shape and location of the parabola, making it possible to solve for unknowns within its equation.

Conic Sections

Conic sections represent the set of curves obtained by intersecting a plane with a double-napped cone. The primary conics are circles, ellipses, parabolas, and hyperbolas. Each has its own unique equation and properties.

Parabolas, such as the one in this exercise, are conic sections notable for their U-shape and the presence of a defining vertex. They are defined by a quadratic equation and are characterized by an axis of symmetry and a focus. In the context of this problem, a parabola opening to the right is formularized by the equation \( y^2 = 4px \). This equation helps in identifying the geometric properties and behaviors of the parabola.

Understanding conic sections allows us to predict how parabolas behave and interact with their environment, be it physically or mathematically. These insights help explain phenomena like the path of projectiles or the reflection properties of satellite dishes, all of which tie back to the geometric characteristics of conic sections.