Problem 59 - College Algebra

Question

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. $$ V(-3,-1), \text { Endpoints }(0,5),(0,-7) $$

Step-by-Step Solution

Answer

The equation of the parabola is \( y = \frac{1}{12}(x + 3)^2 - 1 \).

Step 1: Identify the vertex form of a parabola

The vertex form of a parabola is given by the equation \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. From the exercise, we are given the vertex \((-3, -1)\). Substituting \(h = -3\) and \(k = -1\), we have the equation in the form:\[ y = a(x + 3)^2 - 1 \]

Step 2: Determine the orientation of the parabola

The endpoints of the latus rectum are given as \((0, 5)\) and \((0, -7)\). Since both endpoints have the same x-coordinate and differ in their y-coordinate, the parabola opens vertically (either upward or downward).

Step 3: Calculate the focal length and the value of 'a'

The length of the latus rectum is \(|5 - (-7)| = 12\). For a parabola opening vertically, the focal length, \(f\), is related to the latus rectum by \(4f = 12\). Solving for \(f\) gives \(f = 3\). Therefore, the value of \(a\) in terms of focal distance is \(a = \frac{1}{4f} = \frac{1}{12}\).

Step 4: Formulate the equation of the parabola

Substitute \(a = \frac{1}{12}\) into the vertex form from Step 1:\[ y = \frac{1}{12}(x + 3)^2 - 1 \]Thus, the equation of the parabola is finalized.

Key Concepts

Vertex Form

The vertex form of a parabola is an extremely useful equation for graphing and understanding parabolic shapes. It is written as \[ y = a(x - h)^2 + k \]where

• \((h, k)\) is the vertex of the parabola.
• "a" determines the direction and the width of the parabola.

In this format, you can easily see how the parabola is shifted in the coordinate plane. The vertex, the point \((h, k)\), is a crucial feature because it acts as the "turning point" where the parabola changes direction.
For example, in our exercise, the vertex is at \((-3, -1)\), meaning the parabola turns at this point. By changing the value of "a," the parabola can open wider or more narrow, and can flip upside down.

Latus Rectum

The latus rectum is a special line segment associated with parabolas. It passes through the focus and is perpendicular to the principal axis. This segment is crucial as it helps determine the shape and size of the parabola.
The endpoints of the latus rectum are equidistant from the focus. In the exercise, the endpoints \((0,5)\) and \((0,-7)\) tell us the latus rectum has a length of 12 units.
Understanding its length can help infer other important properties about the parabola, such as the focal length. The latus rectum directly ties to the equation \(4f = l\), where

• \(f\) is the focal length.
• \(l\) is the length of the latus rectum.

Focal Length

The focal length \(f\) of a parabola is the distance between the vertex and the focus. It is an important concept to grasp, as it impacts how "steep" or "flat" the parabolic curve will be. For vertical parabolas, the latus rectum and focal length have a specific relation: \[ 4f = \text{length of latus rectum} \]In our exercise, since the length of the latus rectum is 12, we find the focal length by solving \[ 4f = 12 \] \(f = 3\).
This tells us the focus is 3 units away from the vertex. Additionally, knowing \(f\) helps calculate the parameter "a" in the vertex form: \(a = \frac{1}{4f}\). This aspect directly affects the "spread" of the parabola.

Parabola Orientation

Orientation describes the direction in which the parabola opens.
From the endpoints of the latus rectum in this exercise, both points have the same x-coordinate \((0, 5)\) and \((0, -7)\). This orientation suggests that the parabola is vertical.
For vertical parabolas:

• If "a" is positive, the parabola opens upwards.
• If "a" is negative, the parabola opens downwards.

In our exercise, with \(a = \frac{1}{12}\), the parabola opens upwards.
Knowing the orientation is crucial as it helps predict how the graph will appear based on the vertex form.