Problem 58 - College Algebra with Corequisite Support

Question

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. \(V(1,2),\) Endpoints (-5,5),(7,5)

Step-by-Step Solution

Answer

The equation of the parabola is \((x - 1)^2 = 12(y - 2)\).

Step 1: Identify the Properties of the Parabola

To find the equation of a parabola, we need to identify its properties. We know the vertex of the parabola is at \((1,2)\) and the endpoints of the latus rectum are \((-5,5)\) and \((7,5)\). This means the latus rectum is horizontal, indicating a vertical parabola. The midpoint of the endpoints \((-5,5)\) and \((7,5)\) gives the coordinates of the focus of the parabola.

Step 2: Calculate the Midpoint of the Latus Rectum

The formula for the midpoint of a segment given two endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \(\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)\). Applying this to the endpoints \((-5, 5)\) and \((7, 5)\), we calculate:\[ \text{Midpoint} = \left( \frac{-5 + 7}{2}, \frac{5 + 5}{2} \right) = (1, 5) \]This midpoint \((1, 5)\) is the focus of the parabola.

Step 3: Determine the Direction and Distance from Vertex to Focus

The parabola opens vertically (since the latus rectum is horizontal) and we have the vertex at \((1, 2)\) and the focus at \((1, 5)\). The parabola opens upwards because the focus \((1, 5)\) is above the vertex \((1, 2)\). The distance from the vertex to the focus is \(|5-2| = 3|\).

Step 4: Derive the Equation of the Parabola

The standard form of a vertical parabola opening upwards is \((x - h)^2 = 4p(y - k),\) where \((h, k)\) is the vertex of the parabola and \(p\) is the distance from the vertex to the focus. Given the vertex \((1, 2)\) and the distance \(p = 3\), the equation of the parabola becomes:\[(x - 1)^2 = 4 \times 3 \times (y - 2) \]\((x - 1)^2 = 12(y - 2)\).

Key Concepts

Understanding the Vertex

The vertex of a parabola is a key point and often serves as the "turning point" of the curve. In this particular exercise, the vertex is at \((1, 2)\). This means that the lowest (or highest, depending on direction) point on the parabola is at that location.
The concept of a vertex is important because it helps establish the position of the parabola in the coordinate plane.

• The vertex acts as a reference point for the shape and direction of the parabola.
• It is in the equation of the parabola as \((h, k)\), serving as its center.
• For vertical parabolas, the vertex can help us identify if the parabola opens upwards or downwards.

Understanding the vertex assists in plotting the parabola and determining its overall shape.

Exploring the Focus

The focus of a parabola is another crucial point, essential for fully defining its curve. In the exercise, the focus is determined by finding the midpoint of the endpoints of the latus rectum. This gives us the coordinates \((1, 5)\). The focus sits inside the parabolic curve and is used in defining its shape.

• The distance from the vertex to the focus, denoted as \(p\), helps determine the "spread" or width of the parabola.
• A reflective property means that any ray coming into the parabola parallel to its axis will reflect through the focus.
• Being higher than the vertex \((1, 2)\), it confirms that the parabola is opening upwards.

The focus provides a basis for the formula of the parabola and its overall geometry.

Defining the Latus Rectum

The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry passing through the focus. In our problem, it has endpoints at \((-5, 5)\) and \((7, 5)\). The latus rectum helps in understanding the "width" of the parabola's opening.

• The length of the latus rectum gives a sense of the curvature of the parabola; a longer segment suggests a wider spread.
• Its midpoint is the focus, so knowing the latus rectum allows locating the focus easily.
• In the context of vertical parabolas, a horizontal latus rectum means that it is parallel to the x-axis, reinforcing the shape's symmetry.

Identifying the latus rectum is important for visualizing the parabola's position and size.

Traits of a Vertical Parabola

A vertical parabola is one that opens either upwards or downwards along the y-axis. In this exercise, because the endpoints of the latus rectum are horizontally aligned \((-5, 5)\) and \((7, 5)\), we conclude the parabola is vertical.
This pattern helps in determining how the width of the parabola spreads horizontally. Here is what defines a vertical parabola:

• The vertex form of the equation is \((x - h)^2 = 4p(y - k)\).
• The vertex serves as the \((h, k)\) in the equation.
• Whether the parabola opens up or down depends on the position of the focus relative to the vertex.

In this scenario, the vertical opening indicates symmetry around the vertical line passing through the vertex, with parameters derived from the focus and latus rectum.