Problem 30 - Precalculus with Calculus Previews

Question

Find an equation of parabola that satisfies the given conditions. Focus \((0,-10),\) vertex (0,0)

Step-by-Step Solution

Answer

The parabola equation is \( x^2 = -40y \).

Step 1: Understand Parabola Definitions and Conditions

A parabola can be defined by its vertex and focus. The vertex is the midpoint of the parabola, and the focus is a point inside the parabola that directly affects its direction and "tightness." Since we have the vertex at (0, 0) and the focus at (0, -10), we know the parabola opens downwards along the y-axis.

Step 2: Determine the Directrix

The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and is equidistant from the vertex as the focus. Since the focus is at (0, -10), and the vertex is at (0, 0), the distance between them is 10 units. Therefore, the directrix is a horizontal line at y = 10.

Step 3: Use the Standard Form Equation for a Vertical Parabola

A parabola with a vertical axis has the form \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex, and \( p \) is the distance from the vertex to the focus. Since the vertex is (0, 0) and the focus is (0, -10), \( p = -10 \).Thus, the equation becomes:\[ x^2 = 4(-10)(y - 0) \]which simplifies to:\[ x^2 = -40y \].

Step 4: Verify the Equation Satisfies Given Conditions

To confirm the equation is correct, check the conditions: - The vertex (0,0): Substituting x = 0 and y = 0 into the equation gives 0 = 0, which holds true. - The focus (0,-10): The focus satisfies the distance criteria, with p = -10 already used to derive the equation.

Key Concepts

Vertex and Focus

To understand a parabola, it's essential to know about its vertex and focus. The vertex is the turning point of the parabola, often considered the "midpoint" or "starting point" for its formation. The focus, on the other hand, is a point through which all the parabolic arcs must pass, guiding the shape and direction of the parabola.
For a parabola with a vertex at

• (0, 0)

and a focus at

• (0, -10)

we can determine that the parabola opens downwards along the y-axis. This direction is crucial because it determines the orientation and position of other features like the directrix.

Directrix of a Parabola

The directrix of a parabola is a line that plays a vital role in defining the parabola's axis and symmetry. It is perpendicular to the axis of symmetry and serves as a guide that is equidistant from the vertex as the focus is.
The directrix helps balance the parabolic curve by maintaining a constant distance from any point on the parabola to both the focus and itself.

• For our problem, where the focus is at (0, -10) and the vertex is at (0, 0), the distance between them is 10 units.
• Thus, the directrix is a horizontal line at y = 10.

This reflects the symmetry of the parabola around its axis, being equidistant from the vertex and focus.

Standard Form of a Vertical Parabola

The standard form of a vertical parabola is expressed by the equation:\[(x - h)^2 = 4p(y - k)\]where

• (h, k) represents the vertex
• p is the distance from the vertex to the focus.

In the case of our exercise, since the vertex is

• (0, 0)

and the focus is

• (0, -10)

the value of \(p\) is -10 because the focus is 10 units below the vertex.
This makes the standard form equation for our parabola:\[x^2 = 4(-10)(y - 0)\]which simplifies to:\[x^2 = -40y\]This equation visually represents the direction and position of the parabola, confirming its downward opening orientation as determined by the positioning of the vertex and focus.