Problem 25 - Algebra and Trigonometry

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F(0,2)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the standard form

The equation of a parabola with its vertex at the origin can be written in the form of \[ x^2 = 4py \] where \( p \) is the distance from the vertex to the focus.

Step 2: Identify the value of p

Since the focus is given as \( F(0,2) \), the value of \( p \) is 2 because the focus is 2 units away from the vertex, which means \( p = 2 \).

Step 3: Substitute p into the equation

Substitute \( p = 2 \) into the standard equation \[ x^2 = 4py \]This gives you the equation: \[ x^2 = 4(2)y \] which simplifies to \[ x^2 = 8y \].

Key Concepts

Vertex Form

The vertex form of a parabola makes it easy to identify the parabola's vertex and the direction it opens. The general equation for a parabola in vertex form is: \[ y = a(x - h)^2 + k \]Here, - \( (h, k) \) represents the coordinates of the vertex. - The value \( a \) affects the width and the direction of the parabola opens. If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.When we start with a vertex at the origin, \( (h, k) = (0, 0) \), the equation simplifies to:\[ y = ax^2 \]This is particularly simple as there are no horizontal or vertical shifts. If the equation of a parabola is given in vertex form, transformations like translations, reflections, and stretches are directly noticeable. This simplicity is what often makes the vertex form popular in solving and graphing problems.

Focus of a Parabola

Every parabola has a point called the focus. This point directly influences the shape and positioning of the parabola. The focus, along with the directrix, defines a parabola as the locus of points equidistant from the focus and directrix.When talking about specific rules for parabolas:- If the parabola is vertical, like \[ x^2 = 4py \] the focus is \((0, p)\), meaning it's \( p \) units separated from the vertex along the y-axis.- If the parabola is horizontal, the focus shifts along the x-axis.The parabola in the original exercise has its vertex at the origin \((0, 0)\) and uses the focus \((0, 2)\). Knowing the focus allows the equation of the parabola to express distances to this particular point, ensuring all parabolic properties are intact.

Standard Form of a Parabola

The standard form of a parabola is a fundamental type of equation, making it straightforward to learn the parabola's direction and to calculate its features. It takes the following form when the parabola opens upwards or downwards:\[ x^2 = 4py \]Where:- \( p \) is the distance from the vertex to the focus.In this exercise, since the focus is \( (0, 2) \), \( p = 2 \). By inserting \( p \) into the equation, we find:\[ x^2 = 4(2)y \]Which simplifies to:\[ x^2 = 8y \]This tells us that the parabola opens upward (since \( y \) is positive). The vertex remains at the origin \((0, 0)\), making calculations simpler. The standard form of a parabola is especially useful in analytical geometry, ensuring that concepts such as directrix and axis of symmetry are quickly accessible.