Problem 34 - Algebra and Trigonometry

Question

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(y=6\)

Step-by-Step Solution

Answer

The equation of the parabola is \(x^2 = -12y\).

Step 1: Understand the Components of a Parabola

A parabola is defined by its vertex, focus, and directrix. The vertex is the midpoint between the focus and the directrix line. Given the vertex is at the origin (0,0) and the directrix is at \(y = 6\), we will use these to find the focus.

Step 2: Determine the Focus

The vertex is halfway between the focus and the directrix. The directrix is at \(y = 6\), and since the vertex \((0,0)\) lies on the y-axis, the focus of the parabola, \((0,p)\), will be equidistant from the vertex as the directrix. Thus, the midpoint \(y\) of the vertex and directrix establishes that \((0,p)\) is \(-p = 6\), solving gives \(p = -3\). Therefore, the focus is \((0,-3)\).

Step 3: Use the Parabola Equation

The standard form for a vertical parabola is \(x^2 = 4py\). We know from Step 2 that \(p = -3\). Substitute \(p\) into the equation: \[ x^2 = 4(-3)y \].

Step 4: Simplify the Equation

Substitute \(p = -3\) into \(x^2 = 4py\) and simplify: \[ x^2 = -12y \]. Thus, the equation of the parabola is \(x^2 = -12y\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point that defines its shape and position. In a parabola expressed as either \( y = ax^2 + bx + c \) when open upwards or downwards, or \( x = ay^2 + by + c \) when open sideways, the vertex can be quickly identified as the turning point of the curve.
For parabolas that have equations in the form \( (x-h)^2 = 4p(y-k) \) or \( (y-k)^2 = 4p(x-h) \), the vertex is conveniently located at the point \((h, k)\).

• When a parabola opens upwards or downwards, use \( x = h \) and solve for \( y \) to get the vertex \((h, k)\).
• When a parabola opens sideways, use \( y = k \) and solve for \( x \) to identify the vertex \((h, k)\).

In our specific exercise, the vertex is at the origin \((0, 0)\), simplifying calculations since no shifts are involved. This makes the vertex both the minimum or maximum of the parabola depending on its orientation.

Focus of a Parabola

The focus of a parabola is another point that is fundamentally important in determining the parabola's properties. If you have a vertex at \((h, k)\), the focus is typically some distance \(p\) away from the vertex along the axis of symmetry of the parabola.
The relationship between the vertex \((h, k)\) and the focus \((h, k + p)\) for a vertical parabola is crucial to understanding the parabola's orientation and width. This means that the focus lies either above or below the vertex, depending on whether the parabola opens upwards or downwards.
In our exercise, the vertex is \((0, 0)\) and the calculated focus is \((0, -3)\), highlighting that our parabola opens downwards along the negative \(y\)-axis. The focus being at \((0,-3)\) means every point on the parabola is equidistant from the focus and the directrix \(y=6\).
This information helps construct the equation of the parabola, as you know the direction it opens and how stretched it is, informed by the value of \(p\).

Directrix of a Parabola

The directrix of a parabola is a unique line that helps shape the parabola and lies opposite the focus. For every point \((x, y)\) on the parabola, the distance to the focus is equal to the perpendicular distance to the directrix.
When a parabola has its vertex at \((h, k)\), along with focus and directrix, you can use these components to fully describe it mathematically. The directrix acts as a guideline that is \(p\) units away from the vertex, on the side opposite of the focus, and is often represented by the equation \(y = k - p\), when the parabola opens vertically.

• For vertical parabolas, directrix is a horizontal line: \(y = k - p\).
• For horizontal parabolas, the directrix is a vertical line: \(x = h - p\).

In the provided exercise, the directrix \(y = 6\) means that the parabola is oriented vertically and the distance \(p\) reflects how far the vertex is from the focus and the directrix. This concept helps verify the orientation of the parabola and is used in deriving its equation.