Problem 36 - Precalculus: Mathematics for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus: \(F\left(-\frac{1}{12}, 0\right)\)

Step-by-Step Solution

Answer

The equation is \(y^2 = -\frac{1}{3}x\).

Step 1: Understand the Standard Form of a Parabola

A parabola with its vertex at the origin and a horizontal axis (opening right or left) can be described by the equation \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus. Since the focus given is \(F\left(-\frac{1}{12}, 0\right)\), the parabola opens to the left, meaning \(p\) will be negative.

Step 2: Determine the Distance to the Focus

In our scenario, the vertex is at the origin \((0,0)\) and the focus is at \(\left(-\frac{1}{12}, 0\right)\). The distance \(p\) from the vertex to the focus is \(-\frac{1}{12}\) because the focus is to the left of the vertex.

Step 3: Substitute p into the Parabola Equation

Substitute \(p = -\frac{1}{12}\) into the standard form equation \(y^2 = 4px\):\[y^2 = 4\left(-\frac{1}{12}\right)x\]Simplifying gives:\[ y^2 = -\frac{1}{3}x \]

Step 4: Finalize the Equation

The equation \(y^2 = -\frac{1}{3}x\) describes a parabola with its vertex at the origin \((0,0)\), opening to the left towards the focus \(F\left(-\frac{1}{12}, 0\right)\).

Key Concepts

Vertex form

The vertex form of a parabola is a useful way to gain insights into its transformation and orientation. It is expressed as \( y = a(x - h)^2 + k \), where:

• \( (h, k) \) is the vertex of the parabola.
• \( a \) determines the width and the direction (up/down or right/left) of the parabola.

When the vertex is at the origin \((0, 0)\), the equation simplifies to \( y = ax^2 \) for parabolas opening vertically, or \( x = ay^2 \) for those opening horizontally.
In cases where the vertex and focus are both on the x-axis or y-axis, appropriate orientation and transformations are considered based on the focus' position. For a parabola with a given vertex at the origin, utilizing the vertex form allows us to easily adjust the orientation based on whether the parabola opens to the left, right, upwards, or downwards.

Focus of a parabola

The focus of a parabola is a fixed point used to define and construct the curve itself. The parabola is the locus of points equidistant from a point called the focus and a line called the directrix.

• If the focus is horizontally positioned on the x-axis, it can define a parabola that opens left or right.
• For a focus vertically positioned, the parabola opens upwards or downwards.

In the exercise example, the focus is \( F\left(-\frac{1}{12}, 0\right) \), indicating the parabola opens to the left since the x-coordinate of the focus is negative. This can be seen from the equation \( y^2 = 4px \), where \( p \) defines the direction of the opening, determining how the curve wraps around the focus as it moves further from the vertex.

Distance to focus

The distance from the vertex of a parabola to its focus is crucial for formulating its equation. This distance is represented by \( p \) in the standard form equation \( y^2 = 4px \) for horizontal parabolas or \( x^2 = 4py \) for vertical ones.
In practical terms, \( p \) tells us how far away from the vertex the focus lies:

• When \( p \) is positive, the parabola opens to the right (horizontal) or upwards (vertical).
• When \( p \) is negative, it opens to the left (horizontal) or downwards (vertical).

In the given example, \( p = -\frac{1}{12} \) demonstrates that the parabola opens leftward, indicating that each point on the parabola is set such that the distance to the focus matches the distance to the equivalent point on the directrix, thus shaping the characteristic curve of the parabola.