Problem 16 - Algebra 2

Question

Identify the focus and the directrix of the graph of each equation. $$ y=\frac{1}{4} x^{2} $$

Step-by-Step Solution

Answer

The focus of the given parabola \(y = \frac{1}{4}x ^ 2\) is at the point (0, 1) and the directrix is the line \(y = -1\).

Step 1: Identify the value of 'p'

In the given equation \(y = \frac{1}{4} x^2\), we can see that it's in the form \(y = \frac{1}{4p}x^2\). Hence, by comparison, we find that \(p = 1\).

Step 2: Find the focus

The focus of a parabola \(y = \frac{1}{4p}x^2\) is at the point (0, p). Hence, by substituting p = 1, we come to the conclusion that the focus of the given parabola is at the point (0, 1).

Step 3: Find the directrix

The directrix of a parabola \(y = \frac{1}{4p}x^2\) is the line \(y = -p\). Hence, by substituting p = 1, we come to the conclusion that the directrix of the given parabola is the line \(y = -1\).

Key Concepts

Focus and Directrix

In the study of parabolas, the focus and directrix are crucial features that help to define the curve's geometric properties. A parabola is more than just a mere shape; it has a precise location called the focus and a specific line known as the directrix. These elements uniquely determine the parabola's shape and orientation.

The relationship between a parabola and its focus and directrix is fascinating.

• The focus is a point inside the parabola. Light rays parallel to the axis of symmetry will converge at this focal point.
• The directrix is an external line perpendicular to the axis of symmetry. It helps maintain the shape of the parabola as the locus of points equidistant from both the focus and the directrix.

In the equation provided, the vertex of the parabola is at the origin (0,0). We found that for the equation \(y=\frac{1}{4}x^2\), the focus is at (0,1) and the directrix is the line \(y=-1\). The distance from the vertex to the focus is equal to the distance from the vertex to the directrix, which is known as the focal length, and equates to 'p', in this case, 1.

Quadratic Equations

Quadratic equations form the basis for understanding many types of curves, including parabolas. These equations generally take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. They represent parabolic curves when graphed on a coordinate plane.

For the given exercise, we focus on an equation in a slightly different format, namely \(y = \frac{1}{4}x^2\).

• This equation doesn't have linear or constant terms, which places the vertex at the origin.
• Notice that it is specifically of the form \(y = \frac{1}{4p}x^2\), used for vertical parabolas.
• The absence of terms involving \(x\) or constants simplifies calculations for the vertex, focus, and directrix.

This particular form, known as the standard form for a simple parabola, allows us to easily find the focus and directrix by identifying 'p'. Understanding this helps unravel the inherent symmetry and properties of quadratic functions representing parabolas.

Conic Sections

Parabolas are an integral part of conic sections, which include ellipses, parabolas, and hyperbolas. These are the shapes you get when you slice a cone with a plane at various angles. Parabolas are the section formed when the plane is parallel to the edge of the cone.

Conic sections exhibit unique geometric properties:

• Each conic section has its own standard equation, helping to distinguish it from the others.
• Parabolas specifically have the property of reflecting parallel rays to their focus, making them important in fields like optics and radio engineering.
• The position and orientation of the cutting plane affect the shape of the conic section, offering insight into geometry and physics.

Knowledge of conic sections is fundamental because it not only aids in understanding geometry and algebra but also applies to real-world phenomena such as satellite dishes and headlights. Recognizing the interconnections between the various conic sections deepens your comprehension of mathematical functions and their applications.