Problem 31 - Calculus Early Transcendentals

Question

Find an equation for the conic that satisfies the given conditions. Parabola, vertex \((0,0), \quad\) focus \((0,-2)\)

Step-by-Step Solution

Answer

The equation is \(x^2 = -8y\).

Step 1: Understand the Properties of the Parabola

Since the vertex is at (0,0) and the focus is at (0,-2) , this parabola is vertically oriented and opens downward. The vertex is the midpoint between the focus and the directrix, and since the focus is below the vertex, the directrix is a line above the vertex.

Step 2: Determine Distance to Directrix

The distance from the vertex to the focus is 2 units. Since the vertex is equidistant from both the focus and the directrix, the directrix is 2 units above the vertex, at y = 2.

Step 3: Calculate the Parabola's Equation

To find the equation, use the vertex form of a parabola's equation: $(x - h)^2 = 4p(y - k)$ where (h, k) is the vertex and p is the distance from the vertex to the focus. Substituting (h, k) = (0, 0) and p = -2 (negative because the parabola opens downward), we get: $x^2 = -8y$.

Key Concepts

Vertex

The vertex of a parabola is an extremely important point that helps us understand its geometric properties. It is the "turning point" of a parabola, where the curve changes direction.
For the given problem, the vertex is at the point r(0,0). This is a particularly nice vertex location because having a vertex at the origin simplifies the equation of the parabola.
In general, the vertex form of a parabola can be written as:

• \((x - h)^2 = 4p(y - k), \)

where r\((h, k)\) is the vertex.
For our parabola, which opens downward, the vertex form starts as:

• \((x - 0)^2 = 4(-2)(y - 0)\)

This simplifies the process of finding the equation of the parabola, making calculations straightforward. The vertex gives us an anchor point around which the parabola is structured.

Focus

The focus of a parabola is a special point that influences the curve's shape and orientation. In relation to the vertex, the focus determines where the parabola opens.
For our problem, the focus is located at r(0, -2). This means that our parabola will open downward, as the focus is below the vertex on the coordinate plane.
The distance between the vertex and the focus is denoted as \(p\). For this specific parabola, r\(p = -2\) because the focus is two units below the vertex, indicating the parabola opens in the negative \(y\) direction.

• If \(p\) is positive, the parabola opens upwards.
• If \(p\) is negative, it opens downwards.

Knowing the position of the focus relative to the vertex helps us configure the general structure of the parabola correctly.

Directrix

The directrix of a parabola plays a crucial role in its geometric formation. It is a line, not a point, and every point on the parabola is equidistant from the focus and the directrix.
In our exercise, since the vertex is r(0,0) and the focus is (0,-2), the directrix is a horizontal line parallel to the x-axis.
The distance from the vertex to the focus, which is 2 units, gives us the position of the directrix which is located 2 units in the opposite direction from the focus:

• Directrix: \(y = 2\)

This line is pivotal because the parabola is defined as the set of all points equidistant from the focus and this line. By combining understanding the vertex, focus, and directrix, you can derive the full equation of the parabola and its respective orientation. This balance of points and lines brings the parabola to life on the graph, establishing its unique shape.