Problem 39 - Precalculus with Calculus Previews

Question

Find an equation of parabola that satisfies the given conditions. Vertex \((0,0),\) directrix \(y=-\frac{7}{4}\)

Step-by-Step Solution

Answer

The equation of the parabola is \(x^2 = 7y\).

Step 1: Understanding Parabola Properties

A parabola is a set of all points that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. Since the directrix is horizontal (\(y = -\frac{7}{4}\)), the parabola opens upwards or downwards.

Step 2: Determine the Focus

Since the vertex of the parabola is at \((0,0)\), the distance from the vertex to the directrix is \(\frac{7}{4}\) units. Thus, the focus must be \(\frac{7}{4}\) units from the vertex on the opposite side in the y-direction. Therefore, the focus is at \((0, \frac{7}{4})\).

Step 3: Write the Vertex Form Equation of Parabola

For a parabola with vertex \((h,k)\) and opening upwards or downwards, the vertex form is \((x-h)^2=4p(y-k)\) where\(p\) is the distance from the vertex to the focus. Here, \((h, k) = (0, 0)\) and \(p= \frac{7}{4}\).

Step 4: Plug Values into the Equation

Since the vertex is at \((0,0)\), the equation simplifies to \(x^2 = 4p(y)\). Plugging the value of \(p\), we get \(x^2 = 4\cdot \frac{7}{4} \cdot y\).

Step 5: Simplify the Equation

Simplifying, we get \(x^2 = 7y\). This is the equation of the desired parabola with the given vertex and directrix.

Key Concepts

Vertex Form Equation

The vertex form of a parabola is a helpful way of expressing the equation of a parabola. It highlights the vertex, making it easy to identify the turning point of the parabola. The vertex form equation is written as \[(y - k) = 4p(x - h)^2\] where

• \((h,k)\) is the vertex of the parabola, providing you with the lowest or highest point on the curve, depending on how the parabola opens.
• \(p\) is a value that tells us how "stretched" or "compressed" the parabola is. It represents the distance between the vertex and the focus.

In cases where the parabola opens upwards or downwards, the equation is adjusted to \[(x-h)^2 = 4p(y-k)\].This orientation uses the x-variable squared, aligning with how the parabola direction is set. For instance, in our exercise, the parabola has its vertex at \((0,0)\) and opens upwards. This led to the equation \[x^2 = 7y\], making it simple to see how its shape is defined by these specific conditions.

Focus and Directrix

A parabola's focus and directrix are fundamental elements needed to define its complete structure. The **focus** of a parabola is a fixed, singular point. Every point on the parabola is equidistant from this focus and a straight line known as the **directrix**.
The relationship between the focus and directrix governs the path of the parabola, determining its direction and curvature.

• The **focus** is located "p" units away from the vertex along the line of symmetry, in contrast to the directrix.
• The **directrix** is a line perpendicular to the axis of symmetry of the parabola.

In our given problem, the vertex is at the origin \((0,0)\) and the directrix is at \(y = -\frac{7}{4}\), which reveals that the parabola opens upwards. The focus is found by moving the same distance from the vertex in the opposite direction, confirming it at \((0, \frac{7}{4})\). This calculated position complements the equation by giving a precise balance between the focus and directrix.

Properties of Parabolas

The properties of parabolas are distinctive and intriguing, offering predictability in their geometry and applications. Key properties include:

• A parabola exhibits symmetry about its axis. This means one side mirrors the other.
• It has a well-defined vertex as either its lowest or highest point, depending on its orientation.
• The distance between the points on the curve and the focus and directrix is consistent.

These characteristics make parabolas useful in various fields, such as physics, engineering, and graphics. Understanding properties helps anticipate how the parabola behaves in different scenarios. For this exercise, knowing that \(p = \frac{7}{4}\) allows us to control the spread or squish of the parabola compared to a standard parabola opening along the y-axis. These simple, yet powerful principles are central to mastering parabolas and using them effectively in coursework and practical applications.