Find The Focus Of A Parabola: Equation Explained

Finding the Focus of a Parabola: A Step-by-Step Guide

Understanding the standard $(x, y)$ coordinate plane is crucial when we delve into the fascinating world of conic sections, and parabolas are no exception. When faced with an equation like $12x = (y+4)^2 - 36$, our primary goal is to locate specific features of the parabola, such as its vertex, axis of symmetry, and most importantly, its focus. The focus is a point that defines the shape of the parabola; all points on the parabola are equidistant from the focus and the directrix. Let's break down how to find the focus for the given equation. Our journey begins with recognizing the standard forms of a parabola. A parabola opening horizontally will have an equation in the form $(y-k)^2 = 4p(x-h)$, while a parabola opening vertically will be in the form $(x-h)^2 = 4p(y-k)$. Here, $(h, k)$ represents the coordinates of the vertex, and $p$ is the distance from the vertex to the focus (and also from the vertex to the directrix). The sign of $p$ determines the direction the parabola opens. In our equation, $12x = (y+4)^2 - 36$, we can see a $(y+4)^2$ term, which immediately tells us this parabola opens horizontally. Our next critical step is to rearrange the equation to match the standard form. We want to isolate the squared term and the $x$ term on opposite sides of the equation. To do this, we can add 36 to both sides: $12x + 36 = (y+4)^2$. Now, we need to factor out the coefficient of the $x$ term. In the standard form $(y-k)^2 = 4p(x-h)$, the coefficient of the $x$ term on the right side is $4p$. So, we factor out 12 from the left side: $12(x+3) = (y+4)^2$. To precisely match the standard form, we divide both sides by 12: $(y+4)^2 = 12(x+3)$. Now, we can clearly see our parabola's characteristics. Comparing this to $(y-k)^2 = 4p(x-h)$, we can identify: * The vertex $(h, k)$: By looking at $(y+4)^2$ and $(x+3)$, we can deduce that $k = -4$ and $h = -3$. So, the vertex of our parabola is at $(-3, -4)$. * The value of $4p$: We see that $4p = 12$. This means $p = 12/4 = 3$. Since $p$ is positive, the parabola opens to the right. The focus of a horizontal parabola is located at $(h+p, k)$. Plugging in our values, we get $(-3+3, -4)$, which simplifies to $(0, -4)$. Therefore, the location of the focus of the parabola defined by the equation $12x = (y+4)^2 - 36$ is at $(0, -4)$. This point is fundamental in understanding the parabola's geometric properties and how it reflects or focuses incoming parallel rays. The process of rewriting the equation into a standard form is a key skill in analytic geometry, allowing us to extract vital information about the conic section with relative ease. It's all about recognizing patterns and manipulating the equation systematically. The coordinate plane, with its $x$ and $y$ axes, provides the perfect canvas for visualizing these geometric shapes and their properties. The focus, in particular, is a cornerstone of the parabola's definition, and its location is directly tied to the vertex and the parameter $p$. Mastering these concepts allows us to solve a wide array of problems involving parabolas, from graphing them to understanding their applications in fields like optics and engineering.

Decoding the Equation: A Closer Look at Parabola Properties

Let's delve deeper into the equation of a parabola and what each component signifies, especially for the equation $12x = (y+4)^2 - 36$. As we've established, the standard form for a horizontal parabola is $(y-k)^2 = 4p(x-h)$. Our rearranged equation is $(y+4)^2 = 12(x+3)$. This transformation is not merely a cosmetic change; it's the key to unlocking the parabola's defining characteristics. The vertex, denoted by $(h, k)$, is the