Problem 51 - Precalculus

Question

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (4,3)\(;\) focus: (6,3)

Step-by-Step Solution

Answer

The standard form of the equation for the mentioned parabola is \((y-3)^2 = 8(x-4)\).

Step 1: Identify the vertex and the focus

The vertex, denoted as (h,k), is the point (4,3). The focus, a point denoted as (h+p,k), is given as (6,3).

Step 2: Determine the direction of opening of the parabola

From the vertex and focus values, we can see that focus has a larger x coordinate but the same y coordinate. This indicates that the parabola opens to the right.

Step 3: Find the value of \(p\)

\(p\) is the distance from the vertex to the focus. We can calculate \(p\) by subtracting the x-coordinate of the vertex from the x-coordinate of the focus. Thus, \(p = 6-4 = 2\).

Step 4: Substitute in the standard form equation

Substitute \(h\), \(k\), and \(p\) into the standard form equation: \((y-k)^2 = 4p(x-h)\) to get the final equation. Inserting our values we get: \((y-3)^2 = 4*2(x-4)\).

Step 5: Simplify the equation

Simplifying, we get: \((y-3)^2 = 8(x-4)\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a significant point that can be thought of as the "tip" or the "turning point" of the parabola. This is where the parabola changes direction. It is denoted by

• (h, k),
• where h is the x-coordinate,
• and k is the y-coordinate.

In our exercise, the vertex of the parabola is at the point (4, 3). This means that the parabola shifts to meet the minimum or maximum at this point. Generally, the vertex helps in predicting the behavior of the graph across the coordinate plane.

Focus of a Parabola

The focus of a parabola is a point inside the curve that plays a critical role in its shape and direction. It is at this point that the parabolic shape directs or "focuses" the line of sight towards.

• For the given exercise, the focus is at (6, 3).

The focus is aligned with the x-component of the vertex, impacted by the horizontal axis in this example. The fact that the y-coordinate of the focus matches the vertex y-coordinate, means it provides information about the parabola's direction.

Standard Form of a Parabola

Parabolas have a specific standard equation that helps us express their geometric properties algebraically. For a horizontally-opening parabola, the standard form is given by:

• \[(y - k)^2 = 4p(x - h)\]

Here,

• \(h\) and \(k\) are the vertex coordinates
• \(p\) is the distance from the vertex to the focus

In this exercise, with \(h = 4, k = 3,\) and \(p = 2\), the equation becomes \((y-3)^2 = 8(x-4)\). This expression is crucial for plotting and understanding the orientation of such parabolas on a graph.

Direction of Opening

The direction a parabola opens depends largely on the positioning of the focus relative to the vertex. There are two main types of orientation to consider:

• Vertical Opening: If the coordinates of the focus and vertex differ in the y-axis.
• Horizontal Opening: If the differences are along the x-axis.

In our example, since the vertex (4, 3) and the focus (6, 3) share the same y-coordinate, the parabola opens horizontally to the right. This opening direction is controlled by having the focus to the right of the vertex.

Distance from the Vertex to the Focus

The distance from the vertex to the focus is an important parameter represented by \(p\). This distance helps in determining the shape of the parabola equation. It translates algebraically to the space between these two points. For finding \(p\), simply calculate the difference between the coordinates:

• Here, \(p = |6 - 4| = 2\) since we only consider the x-coordinates due to horizontal alignment.

Understanding this distance helps ensure precise graphing of the parabola, allowing one to determine how "wide" or "narrow" the parabola appears on the planar surface.