Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the parabola given information about its graph. Vertex is \((\sqrt{2},-\sqrt{3}) ;\) directrix is \(x=2 \sqrt{2},\) focus is \((0,-\sqrt{3})\).

Step-by-Step Solution

Answer

The equation of the parabola is \((y + \sqrt{3})^2 = -4\sqrt{2}(x - \sqrt{2})\)."

Step 1: Identify the Orientation of the Parabola

Since both the vertex and the focus have the same y-coordinate, \(y = -\sqrt{3},\) the parabola is horizontal. The vertex is \( (\sqrt{2}, -\sqrt{3}) \), and the axis is aligned along the x-axis.

Step 2: Use the Formula for a Horizontal Parabola

For a horizontal parabola, the equation is \((y - k)^2 = 4p(x - h)\), where \( (h, k) \) is the vertex, and \( p \) is the distance from the vertex to the focus or directrix.

Step 3: Calculate the Value of \(p\)

The vertex is \( \sqrt{2} \), and the directrix is \( x = 2\sqrt{2} \). The distance between the vertex \((h, v)\) and the directrix is \(2\sqrt{2} - \sqrt{2} = \sqrt{2}\), hence \( p = -\sqrt{2} \), indicating the parabola opens to the left (as the focus is closer to the vertex than the directrix).

Step 4: Write the Equation of the Parabola

Plug the values into the parabola's equation: \[(y + \sqrt{3})^2 = -4\sqrt{2}(x - \sqrt{2})\] This equation represents a horizontally opening parabola with vertex at \( (\sqrt{2}, -\sqrt{3}) \) and directrix \( x = 2\sqrt{2} \).

Key Concepts

Vertex of the Parabola

The vertex of a parabola is a critical point where the curve changes its direction. Think of it as the parabolic equivalent of the center. For a horizontally oriented parabola, like in our exercise, the vertex is the rightmost or leftmost point on the curve. It's the peak if the parabola opens sideways.
For this particular example, the vertex is located at \((\sqrt{2}, -\sqrt{3})\). This means that the parabola's central point, with respect to x-axis and y-axis movement, is precisely here.
Understanding the vertex is essential because it is used in the parabola's equation \((y - k)^2 = 4p(x - h)\) for horizontally oriented parabolas. Here, \((h, k)\) is the vertex, influencing where the parabola starts or peaks.

Directrix of Parabola

The directrix of a parabola is a straight line that helps define the curve. It runs parallel to the axis of symmetry and has a unique relationship with the focus. For a horizontal parabola, the directrix is vertical, while for a vertical parabola, this line is horizontal.
In this scenario, our directrix is given by \(x = 2\sqrt{2}\). This line lies alongside the x-axis, establishing where the parabola's curve gets its symmetric stretch. The distance from any point on the parabola to the focus is equal to the distance from that point to this directrix, reflecting a fundamental property of parabolas.
Knowing the directrix allows us to understand the parabola's width and direction, contributing to calculating the "p" value in the equation of the parabola.

Focus of the Parabola

The focus of a parabola is a fixed point that, along with the directrix, determines the entire shape of the parabola. For any point on the parabola, the distance to the focus is equal to the distance to the directrix, an important geometric property.
In this example, the focus is located at \((0, -\sqrt{3})\). This point does not lie on the parabola itself but inside its curve when drawn. Essentially, the parabola wraps around this focus.
Understanding the location of the focus is integral to determining the "p" value, which in turn helps to write the parabolic equation correctly. In our case, it indicates the parabola opens to the left, signifying a negative p value in the equation \((y + \sqrt{3})^2 = -4\sqrt{2}(x - \sqrt{2})\).