Problem 48 - Precalculus: Mathematics for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focal diameter 8 and focus on the negative \(y\) -axis

Step-by-Step Solution

Answer

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

Step 1: Identify parabola orientation

The problem states that the focus is on the negative \(y\)-axis. This indicates that the parabola opens downwards. A downward-opening parabola with its vertex at the origin \((0,0)\) has the general equation \(x^2 = -4py\).

Step 2: Understand focal diameter

The focal diameter of a parabola is given by \(4p\). Since the focal diameter is 8, we equate \(4p = 8\) to find \(p\).

Step 3: Solve for \(p\)

Solve the equation \(4p = 8\) to find \(p = 2\).

Step 4: Write the parabola equation

Substitute \(p = 2\) into the equation for a downward-opening parabola: \(x^2 = -4py\). This gives \(x^2 = -4(2)y\), or \(x^2 = -8y\).

Key Concepts

Vertex

The vertex of a parabola is a pivotal point where the curve changes direction. In this problem, the vertex is located at the origin, (0,0). This serves as the centerpiece of our parabola. When given the vertex, we can easily determine other parameters of the parabola, such as its direction and exact form. With our vertex known, we are equipped to start constructing the parabola equation. The vertex not only indicates where the parabola starts turning but also acts as the reference point for measuring the curvature and size of the parabola.

Focal Diameter

The focal diameter of a parabola, also known as the latus rectum, is the segment that passes through the focus and is perpendicular to the axis of symmetry. It provides insight into the parabola's shape. This line is crucial because it defines the width of the parabola at the point where it intersects with the focus.

• For this problem, the focal diameter is given as 8.
• It is related to the parameter \( p \) by the formula \( 4p \).

The relationship helps to solve for \( p \), which then plugs into the parabola equation, giving us \( p = 2 \). The focal diameter essentially facilitates navigating the curve's spread.

Downward-Opening Parabola

A downward-opening parabola has a specific orientation where the arms of the curve direct downwards. Its orientation is determined by the position of the focus relative to the vertex. In this case, since the focus is on the negative y-axis, our parabola will open downwards. The equation for a downward-opening parabola with its vertex at the origin is \( x^2 = -4py \).

• Here, the negative sign in the equation signifies the downward direction.
• This form helps to easily identify the parabola's opening direction just by looking at the sign.

The equation form allows for quick predictions of how the parabola behaves on a graph.

Focus on y-axis

Locating the focus on the y-axis deeply influences the equation of a parabola. The focus is a fixed point used in the definition of a parabola, from which distances are measured to derive the curve itself. For this exercise, the focus is on the negative y-axis, which not only signals a downward-opening parabola but also helps in affirming the value found for \( p \).

• The position of the focus in relation to the vertex is critical as it tells us the curve's steepness and opening.
• The focus for this parabola is at the point \( (0, -2) \).

This arrangement is essential as it completes the structural framework of our parabola and allows us to write the complete equation.