Problem 45 - Precalculus: Mathematics for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Opens downward with focus 10 units away from the vertex

Step-by-Step Solution

Answer

The equation of the parabola is \(x^2 = 40y\).

Step 1: Identify the Type and Equation of Parabola

Since the parabola opens downward and its vertex is at the origin, the equation will be of the form \(x^2 = -4py\), where \(p\) is the distance from the vertex to the focus. Because the focus is 10 units away, we will use this information to find \(p\).

Step 2: Calculate the Value of p

The given condition states that the focus is 10 units away from the vertex. Because the parabola opens downward, \(p\) will be negative, specifically, \(p = -10\). This means that the distance from the vertex to the focus in the downward direction is 10.

Step 3: Substitute p into the Parabola Equation

Substitute the value of \(p\) into the standard form \(x^2 = -4py\). Here, \(p = -10\), so the equation becomes \(x^2 = -4(-10)y\).

Step 4: Simplify the Equation

Simplify \(x^2 = 40y\) by performing the multiplication to get the final equation of the parabola.

Key Concepts

Vertex Form

Understanding the vertex form of a parabola's equation is crucial, especially when dealing with transformations and shifts of the graph. The vertex form is given by the equation \[ y = a(x - h)^2 + k \] where \( (h, k) \) represents the vertex of the parabola. In this specific form, the vertex is easily identifiable, allowing us to understand exactly where the parabola sits on the coordinate plane.
For example, if the parabola's vertex is at the origin, \( (0, 0) \), this simplifies to \[ y = ax^2 \], showing us that the graph has no horizontal or vertical shifts. This form aids us in quickly sketching the parabola and predicting its path. Furthermore, the value of \( a \) determines the parabola's direction and width: if \( a > 0 \), it opens upward, while a negative \( a \) signifies it opens downward.

Focus and Directrix

The focus and directrix are integral components of understanding the geometric definition of a parabola. The focus is a fixed point, and the directrix is a fixed line. Together, they define the parabola as the set of all points equidistant from the focus and the directrix.
In mathematical terms, if a parabola has its vertex at the origin and opens vertically, it's easier to see these relationships through its standard equation. For a parabola that opens downward, such as \[ x^2 = -4py \], the focus lies at \( (0, -p) \). In our exercise, where \( p = -10 \), the focus is at \( (0, 10) \).
The directrix, in this situation, would be a horizontal line given by \( y = -p \). Thus, it would be the line \( y = -10 \). Understanding the placement of the focus and directrix helps in determining the direction and shape of the parabola, making it easier to predict how the graph will look.

Conic Sections

Conic sections arise from the intersection of a plane and a double-napped cone. They include different forms: circles, ellipses, parabolas, and hyperbolas, all distinguished by the angle and position of the intersecting plane.
Parabolas, like the one in our exercise, are formed when the plane is parallel to the cone's side. Each conic section has unique properties and equations that help us understand their graph shapes. For parabolas specifically, knowing how they relate to other conics enhances comprehension of their behavior and uses in various fields, from physics to engineering.
For example, parabolic shapes are critical in reflector systems, like satellites and flashlights, because they focus energy to or from the focal point efficiently. Recognizing the distinct qualities of conic sections can deepen our understanding of both theoretical and real-world applications.