Problem 79 - A Graphical Approach to Precalculus with Limits

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 is \( y^2 = 4x \).

Step 1: Identify the Parabola Form

Since the parabola opens to the right, we use the equation form for a horizontally opening parabola: \( y^2 = 4px \). The vertex is at the origin: (0,0).

Step 2: Substitute the Given Point

Use the coordinates of the given point \((2, -2\sqrt{2})\) to substitute into the equation \( y^2 = 4px \). This substitution gives \((-2\sqrt{2})^2 = 4p(2)\).

Step 3: Simplify the Equation

Calculate \((-2\sqrt{2})^2\), which equals \(8\). Hence, \(8 = 4p(2)\).

Step 4: Solve for \( p \)

We simplify the equation \(8 = 8p\). Divide both sides by 8 to get \(p = 1\).

Step 5: Write the Final Equation

Insert \(p = 1\) back into the parabola's equation form: \(y^2 = 4(1)x\) or \(y^2 = 4x\).

Key Concepts

Vertex

The concept of a vertex is vital when dealing with parabolas. In simple terms, a vertex is the highest or lowest point on a parabola, depending on how it opens. It serves as a kind of "turning point" where the direction of the parabolic curve changes. For a parabola with a vertex at the origin, this point is located exactly at the coordinate (0,0).

Understanding the vertex is crucial because it acts as a reference point to determine the placement of the parabola in the coordinate plane. It also plays a central role in the equation of the parabola, impacting its shape and orientation. When a problem specifies that a parabola's vertex is at the origin, this simplifies your equation tremendously since the terms for the vertex in the equation will be zero.

Equation of a Parabola

The equation of a parabola can take various forms based on its orientation and position. For a simple parabola centered at the origin, the equation depends on whether it opens vertically or horizontally. If the parabola opens horizontally, the classic form is:

• \(y^2 = 4px\) for right or left opening
• \(p\) represents the distance from the vertex to the focus

This distance \(p\) greatly influences the "width" and the orientation of the parabola.

In our problem, since the parabola is horizontally opening with a vertex at the origin, we apply the equation \(y^2 = 4px\). If "p" is positive, the parabola opens to the right, whereas a negative value would indicate it opens to the left. By substituting a point through which the parabola passes, such as \(2, -2\sqrt{2}\), we can calculate the exact value of \(p\), allowing us to write the specific equation for the parabola in question.

Horizontally Opening Parabola

Parabolas can open in different directions, and understanding their orientation is key. A horizontally opening parabola is characterized by its left-to-right or right-to-left opening. In contrast to the more common vertically opening parabolas, the standard form for a horizontally oriented one is \(y^2 = 4px\), rather than \(x^2 = 4py\).

When determining the specific equation for our problem, we first recognize that the parabola opens to the right. This is determined by substituting a given point, through which the parabola passes, into the horizontal form equation. This gives us a means to calculate \(p\). In our scenario, solving the equation with the provided point \((2, -2 \sqrt{2})\) helps ascertain that \(p = 1\), confirming a rightward opening with the equation \(y^2 = 4x\).

Therefore, understanding this orientation not only helps in formulating the correct equation but also visualizing how a parabola will "shape" within a coordinate plane. The horizontal orientation is less common but still vital in the analysis of parabolic paths.