Problem 12 - Algebra and Trigonometry

Question

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$ x^{2}=-20 y $$

Step-by-Step Solution

Answer

The focus of the parabola is at (0, -5) and the directrix is the line y = 5.

Step 1: Identify the Form

The given equation is in the form \(x^{2} = 4py\), but with a negative sign. That inform us the parabola opens downwards.

Step 2: Find the value of p

The coefficient of y in the given equation represents -4p. Therefore we have -4p = -20. Solving for p, we get \(p=5\).

Step 3: Identify focus and directrix

Now, using the value of p=5, we can easily find that the focus of the parabola is \(S=(0,-5)\), as the p is negative. The directrix of the parabola, given \(y=p\), is \(y=-(-5)\), which simplifies to \(y=5\).

Step 4: Graph the parabola

We plot the vertex, focus and directrix on the graph. We first plot the vertex at the origin (0,0), then we plot the focus at \(S=(0,-5)\), and then plot the directrix at \(y=5\). Then, sketch the parabola symmetrically around the axis passing through the vertex and the focus, it opens downwards as indicated by the negative sign, and tangential to the line \(y=5\).

Key Concepts

Understanding the Focus of a Parabola

A parabola is a unique type of curve that is primarily defined by certain characteristic points and lines, which include the focus and directrix. The focus is a point that lies inside the parabola. For a given parabola, each point on the curve is equidistant from this focus and a corresponding line known as the directrix.
This distinct property is key to understanding what makes a parabola’s shape so special.For the equation \(x^2 = -20y\), we know that the form \(x^2 = 4py\) helps identify the nature of the parabola. Here, the focus is determined by finding \(p\) from the equation \(-4p = -20\), resulting in \(p = 5\). Since the parabola opens downward due to the negative sign, the focus is located at the point \(S = (0, -5)\) along the axis of symmetry.
The focus acts as a magnet, influencing the curvature of the parabola and its direction.

Deciphering the Directrix

The directrix of a parabola is another crucial geometric feature. This horizontal line helps define the shape and position of the parabola, and all points on the parabola are equidistant from the directrix and the focus, which is called the reflective property.
The directrix serves as an imaginary guideline that works with the focus to shape the parabola.In the example equation \(x^2 = -20y\), the directrix is found using the formula \(y = -p\). With our calculated value of \(p = 5\), the directrix is the line \(y = 5\).
This line is positioned parallel to the x-axis, opposite from the focus, at an equal distance on the other side of the origin. The interplay between the focus and directrix defines how the parabola opens and extends.

Graphing Parabolas Simplified

Graphing a parabola might seem daunting at first, but it becomes much simpler when you break it down into steps. Begin by understanding the general orientation from the equation. In \(x^2 = -20y\), the negative sign indicates the parabola opens downward.Here’s a step-by-step approach to graphing:

• Locate the vertex, which in standard form is at the origin \((0,0)\).
• Plot the focus \(S=(0, -5)\) found earlier next to the vertex.
• Draw the directrix \(y=5\) as a horizontal line above the vertex.

Next, sketch the parabola. It should curve downward from the vertex, symmetrically around the vertical line through the vertex and the focus. The points on the curve are equidistant from the focus and the directrix.
Emphasizing symmetry and the distance between focus and directrix ensures a balanced, accurate graph.