Problem 31 - College Algebra with Modeling and Visualization

Question

Find an equation of the parabola with vertex \((0,0)\) that satisfies the given conditions. $$ \text { Focus }\left(0, \frac{3}{4}\right) $$

Step-by-Step Solution

Answer

The equation of the parabola is \( y = \frac{1}{3}x^2 \).

Step 1: Understand the Definition of a Parabola

A parabola is a set of all points in the plane that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. In this case, the vertex of the parabola is at the origin \((0,0)\).

Step 2: Identify the Focus and the Vertex

The focus of the parabola given is \(\left(0, \frac{3}{4}\right)\) and the vertex is \((0,0)\). Since the focus is above the vertex, this is a vertical parabola that opens upwards.

Step 3: Determine the Distance from Focus to Vertex

The distance from the vertex \((0,0)\) to the focus \(\left(0, \frac{3}{4}\right)\) is \(p = \frac{3}{4}\). For a parabola with a vertical axis of symmetry and vertex at the origin, the value \(p\) represents the distance to the focus as well as the negative of the distance to the directrix.

Step 4: Write the Equation of the Parabola

For a parabola with vertex \((0, 0)\) and vertical axis of symmetry, the standard form of the equation is: \[ y = \frac{1}{4p}x^2 \]Substituting \(p = \frac{3}{4}\) gives:\[ y = \frac{1}{3}x^2 \].

Step 5: Confirm the Equation Using the Focus

Verify the equation by ensuring that the focus \(\left(0, \frac{3}{4}\right)\) is correct for the equation found. The derived equation has vertex \((0,0)\) and the correct focus for \(p = \frac{3}{4}\), confirming that the equation of the parabola is correct.

Key Concepts

Vertex Form of Parabola

The vertex form of a parabola is a convenient way to express its equation, especially when you know the vertex of the parabola. This form is written as: \[ y = a(x - h)^2 + k \]

• The variable \( a \) dictates the width and the direction of the opening of the parabola.
• The vertex of the parabola is represented by the point \((h, k)\).

For a parabola with a vertex at the origin, \((0,0)\), the vertex form simplifies to \[ y = ax^2 \].This form makes it easy to identify the parabola's vertex, which is useful for graphing and understanding its orientation. In cases where the parabola’s axis of symmetry is vertical, like in our problem, the equation remains simple, allowing for easy substitution of known values, like the focus, to solve for specific parameters.

Focus and Directrix

The focus and directrix are fundamental elements in understanding a parabola's shape and position. These elements help define the set of points that make up the parabola.

• The focus is a fixed point inside the parabola. In our example, it is located at \( (0, \frac{3}{4}) \).
• The directrix is a fixed line outside the parabola, parallel to the base, opposite the opening.

The parabola is always found equidistant between its focus and directrix. For example, the distance from any point on the parabola to the focus is equal to the perpendicular distance from that point to the directrix. This unique property aids in deriving the parabola’s equation.For vertical parabolas, with a vertex at the origin and the focus located at \((0, p)\), the directrix is a horizontal line given by the equation \( y = -p \). Understanding these components is key to solving problems around parabolic shapes and their equations.

Vertical Parabola

A vertical parabola is one where the axis of symmetry, or the line that vertically divides the parabola into two mirror images, is vertical. This structure is characterized by the vertex being directly above or below the focus, depending on the orientation. When the vertex is at the origin, the focus being above the vertex implies that the parabola opens upwards, as in our exercise with the focus at \( \left(0, \frac{3}{4}\right) \). The standard equation for a vertical parabola with a vertex at the origin is:\[ y = \frac{1}{4p}x^2 \]The distance \( p \) from the vertex to the focus helps define curvature and direction. A positive \( p \) makes it open upwards, while a negative \( p \) would make it open downwards, reflecting a different set of characteristics.

Conic Sections

Parabolas are a type of conic section, a group of curves formed by the intersection of a right circular cone and a plane. Conic sections include parabolas, circles, ellipses, and hyperbolas.

• Parabolas are unique because they are formed when the plane is parallel to a generatrix line of the cone.
• This orientation means there's only one continuous curve, distinct from the closed formation of circles and ellipses or the two separate curves of hyperbolas.

Understanding parabolas as conic sections helps in comprehending their geometric properties and behaviors in various contexts such as physics for projectile motion or engineering in structural designs. The standard form equations and the shapes of these curves hold significance across various fields, underscoring the importance of recognizing their foundational mathematics.