Problem 19 - Precalculus

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: (-2,0)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(y^2 = -8x\).

Step 1: Identify the Focus and the Vertex

In the given question, the vertex of the parabola is given at origin, (0,0), and the focus is given as (-2,0). The parabola is oriented to the left because the focus is to the left of the vertex.

Step 2: Determine the value of p

The value of p is the distance from the vertex to the focus along the axis of the parabola. Since the vertex is at origin (0,0) and focus is (-2,0), the p-value is -2 because the focus is to the left of the origin on the x-axis.

Step 3: Write the equation using standard form

Now by substituting p = -2 into the standard equation for the parabola that opens to the left or right, \(y^2 = 4px\), the equation becomes \(y^2 = -8x\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is an essential point that represents the highest or lowest point on the curve, depending on its orientation. It is where the parabola changes direction. For parabolas that open upward or downward, the vertex is the minimum or maximum point, respectively. Similarly, for parabolas that open left or right, the vertex serves as a central point, neither minimum nor maximum, but crucial nonetheless.
In this particular exercise, the parabola’s vertex is located at the origin, which is the point (0,0). This location tells us that the parabola is centered at the origin, simplifying calculations. Knowing the vertex helps us understand the parabola's positioning in the coordinate plane and serves as a reference when plotting its path.

Focus of a Parabola

The focus of a parabola is a fixed point that, along with the directrix, helps in defining the curve. Every point on the parabola is equidistant to the focus and the directrix, giving the parabola its unique shape. The focus lies on the axis of symmetry of the parabola, which is a line that vertically or horizontally divides the parabola into two mirror images.
For the given exercise, the focus is at (-2,0), indicating the parabola opens sideways. Since the focus is to the left of the vertex (0,0), this particular parabola opens to the left. The distance from the vertex to the focus is key in finding the parameter "p," which is used in the standard form equation of the parabola.

Standard Form of a Parabola

The standard form of a parabola's equation depends on the direction in which it opens. For parabolas opening to the left or right, the equation is typically written as \(y^2 = 4px\), where "p" represents the directional distance from the vertex to the focus. When "p" is positive, the parabola opens to the right, while a negative "p" indicates it opens to the left.
In the exercise discussed, since the focus is at (-2,0) and the vertex at the origin (0,0), "p" is -2. Thus, plugging this into our standard form gives \(y^2 = 4(-2)x\), simplifying to \(y^2 = -8x\). This equation gives us all the information needed about the parabola's shape and direction.

Equation of a Parabola

The equation of a parabola provides a mathematical description of its shape and direction. It allows us to determine the path of the curve and its orientation in the coordinate plane. The standard form, either \(y^2 = 4px\) or \(x^2 = 4py\), helps identify where the parabola opens and how wide it appears.
For the parabola opening to the left as addressed in the exercise, the equation \(y^2 = -8x\) was derived using the focus of (-2,0) and the vertex of (0,0). By adjusting the value of "p" and other parameters, you can easily describe parabolas in various positions and scales, making the equation a versatile tool in mathematics.