Problem 87 - A Graphical Approach to College Algebra

Question

Find an equation of a parabola that satisfies the given conditions. Focus \((0,0) ;\) directrix \(x=-2\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y^2 = 4(x+1)\).

Step 1: Understanding the Conditions

The focus of the parabola is given as \((0,0)\) and its directrix as \(x = -2\). Remember that a parabola is the set of all points equidistant from the focus and the directrix.

Step 2: Using the Parabola Definition

The general equation for any parabola given a focus \((h,k)\) and a directrix \(x = d\) is \((y-k)^2 = 4p(x-h)\), where \(p\) is the distance from the vertex to the focus. Here, the focus \((0,0)\) and directrix \(x=-2\) imply the vertex is at \((-1,0)\) (midpoint).

Step 3: Finding the Distance p

The distance \(p\) from the vertex \((-1,0)\) to the focus is 1, since \(0 - (-1) = 1\). Since the parabola opens to the right, \(p = 1\).

Step 4: Forming the Parabola Equation

Substitute \(h = -1\), \(k = 0\), and \(p = 1\) into the parabola equation \((y-k)^2 = 4p(x-h)\). This becomes \(y^2 = 4(x+1)\).

Key Concepts

Focus and Directrix

In the world of conic sections, a parabola stands out because of its unique structure defined by two key elements: the focus and the directrix. The focus is a fixed point inside the parabola, where all points on the curve are equally distant from this point and the directrix. The directrix, in contrast, is a straight line outside of the parabola.

Think of the focus and directrix as the parabolic GPS. They help us pinpoint the location of any point on the parabola. Here's a quick guide on how they work:

• For any point on the parabola, the distance to the focus is the same as the distance to the directrix.
• The parabola is mostly symmetrical around a vertical or horizontal line, called the axis of symmetry, passing through the focus and perpendicular to the directrix.

In this particular problem, our focus is at (0,0), and the directrix is at the line x = -2. This setup means the parabola will open sideways, equidistantly between these elements.

Parabola Properties

Parabolas show distinctive properties based on their equation and orientation. A parabola can face upwards, downwards, or sideways. The orientation depends on the position of the focus and directrix. In mathematical terms, a parabola can be expressed in a standard equation form based on its particular properties.

When dealing with a focus at (0,0) and directrix as x = -2, as in this problem:

• Since the directrix is vertical, the parabola is horizontal.
• The vertex, which is midway between the focus and directrix, is located at (-1,0).
• The parabola opens to the right because the focus is to the right of the directrix.
• The equation of this parabola can be expressed as (y-k)^2 = 4p(x-h), where (h,k) is the vertex.

This results in the equation y^2 = 4(x+1), manifesting the parabola's fundamental attributes including its direction and symmetry.

Conic Sections

Conic sections are fascinating curves obtained by slicing a cone with a plane. Depending on the angle and position of this slice, we get different conic shapes like circles, ellipses, hyperbolas, and notably, parabolas.

Here’s why parabolas are intriguing as conic sections:

• A parabola represents a special case where the slicing plane is parallel to another plane that is tangent to the cone.
• Every parabola is defined using a focus and directrix, making it distinctively Archimedean.
• Parabolas are common in physics and engineering, appearing in the paths of projectiles and the design of reflective surfaces.

Understanding conic sections helps us see the blend of geometry and algebra used to describe natural phenomena and man-made structures. In essence, the parabola from this problem beautifully showcases the interconnection of these sections, defined perfectly by its focus, directrix, and the derived equation.