Problem 20 - Precalculus

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: (0,-2)

Step-by-Step Solution

Answer

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

Step 1: Identify the Given Information

The focus is at (0, -2) and the vertex is at the origin (0, 0). The distance from the vertex to the focus can be determined by the absolute value of the difference in their y-coordinates.

Step 2: Calculate the distance to the focus

Calculate the distance from the vertex to the focus. It is equal to the absolute difference in their y-coordinates, which here gives us a value of 2 (i.e. \(|-2 - 0|\ = 2\)). As the parabola is facing downwards, the value of \(A\) is to be considered negative. Thus, \(A = -2\).

Step 3: Write the Equation of the Parabola

Substitute the calculated value of \(A\) into the standard equation for a downwards facing parabola, which is \(y = -Ax^2\). The equation of the parabola is therefore \(y = -2x^2\).

Key Concepts

Standard Form of a Parabola

Parabolas are unique curves that are often represented by a simple equation, called the standard form. For a parabola oriented in a vertical direction, such as one that opens upwards or downwards, the standard form of its equation is given by:

• Opening upwards: \[ y = ax^2 + bx + c \]
• Opening downwards: \[ y = -ax^2 + bx + c \]

Here, the letters \(a\), \(b\), and \(c\) are coefficients that define the specific shape and direction of the parabola. In the case of our specific problem, where the parabola has a vertex at the origin and opens downwards, this simplifies to \(y = -Ax^2\). The term \(-Ax^2\) indicates the parabola opens downwards, as derived from the given focus. Understanding this form helps in quickly identifying the key properties of a parabola, such as its direction and steepness.

Vertex

The vertex of a parabola is a crucial point where the curve changes direction. For many parabolas, particularly those defined in standard forms starting with \(y\) or \(x\), the vertex represents either the maximum or minimum point. In this exercise, the vertex is located at the origin, or point (0, 0).

• This simplifies many calculations, as both \(x\) and \(y\)-coordinates are zero.
• The vertex serves as a symmetry point, meaning one side of the parabola is a mirror image of the other.

In our situation, the parabola opens downwards from the vertex. The vertex being at the origin also implies that there are no horizontal or vertical shifts in its position, making it quite straightforward to work with.

Focus

The focus of a parabola is a fixed point located within its curve, affecting its shape and the way it reflects light or objects. For a parabolic dish or mirror, for instance, rays parallel to the axis of symmetry pass through this point upon reflection.

• In our example, the focus is at the point (0, -2).
• The distance between the vertex and the focus helps determine important properties of the parabola, such as its depth and width.

The object here is to ensure that the focus lies exactly along the parabola's axis of symmetry, influencing its direction and form. The relative position of the focus below the vertex confirms the downward opening of the parabola.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two identical halves. It runs through the vertex and is central to the parabola's balanced shape.

• For our current parabola, the axis of symmetry is the line \(x=0\).
• This line ensures that for every point on the parabola on one side of the axis, there is a corresponding point at an equal distance on the opposite side.

An understanding of the axis of symmetry aids in sketching the parabola accurately and analyzing its reflective properties. It reinforces the reason for the equation being simple to handle, as the slope gradients on both sides of this line are mirror images. For this exercise, it emphasizes how the parabola's orientation is aligned around its central axis, ensuring symmetry and uniformity in its structure.