Problem 21 - Precalculus with Limits

Question

In Exercises 19-32, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: \((0, \frac{1}{2})\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola with the given characteristics is \(y = 0.5*x^2\)

Step 1: Identify the focus and vertex

The problem states that the vertex is at the origin, so it is (0,0). The focus is given as \((0, \frac{1}{2})\) meaning it is vertically 1/2 units away from the vertex.

Step 2: Calculate the value of 'p'

\'p' is the distance from the focus to the vertex. Because our vertex is at the origin, \'p' would be the y-coordinate of the focus point which is \(\frac{1}{2}\).

Step 3: Calculate the value of 'a' and write the standard form of the equation

Since 'p' is \(\frac{1}{2}\), we can substitute this into the formula 'a' = \(1/4p\) to get 'a' = 0.5. Substituting this value into the general equation \(y = a*x^2\), our equation becomes \(y = 0.5*x^2\)

Key Concepts

Vertex

The vertex of a parabola is a pivotal point in understanding its shape and position in a coordinate system. Think of the vertex as the tip or bottom-most point of the parabola, from which it opens either upwards or downwards. In the problem mentioned, the vertex is at the origin, or the point (0,0). When a parabola's vertex is at the origin, calculations become simpler, because you can rely heavily on symmetry. The vertex acts as a reference point from which to measure other parts of the parabola, such as its focus.

Focus

The focus of a parabola is a fixed point that, together with the directrix, helps in defining the parabola uniquely. The focus lies inside the curve of the parabola. In the exercise provided, the focus is located at

• (0, \(\frac{1}{2}\))

, which tells us that the parabola opens vertically. The focus is always the same distance from the vertex as it is from the point where directrix would be. Calculating this distance is important to find other properties like the focal width, and it is denoted by 'p'. For this parabola, 'p' is \(\frac{1}{2}\), the y-coordinate of the focus.

Standard Form

The standard form of a parabolic equation lets us know its graphical representation with utmost clarity. In general, the standard form when the parabola opens vertically (and vertex at the origin) is written as:

• \(y = ax^2\) , for vertical opening

where 'a' represents the parabola's direction and how "stretched" it is. In the exercise, we calculated 'a' as 0.5, leading us to the equation:

• \(y = 0.5x^2\)

Having this standard form helps not just in graphing but also in understanding how the parabola behaves under various transformations, like shifting the vertex.

Distance Formula

The distance formula is an essential tool in geometry and algebra, allowing us to find the distance between two points in a coordinate plane. It is outlined as follows:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

To understand this in the context of the parabola, consider the distance from the vertex to the focus, which helps calculate 'p'. Here, because the vertex is at the origin and the focus has coordinates (0, \(\frac{1}{2}\)), the distance formula simplifies to calculating the absolute difference in their y-coordinates, which yields:

• \(d = \left| \frac{1}{2} - 0 \right| = \frac{1}{2}\)

This distance is critical for formulating the parabola's equation and understanding its architecture.