Problem 18 - Precalculus

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: \(\left(-\frac{3}{2}, 0\right)\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(x=-\frac{1}{3}y^2\).

Step 1: Determine the direction

Determine the direction in which the parabola opens. Since the x-coordinate of the focus is negative, the parabola opens to the left.

Step 2: Calculate the value of a

The value of a can be found by taking the half of the reciprocal of the x-coordinate of the focus. Since the x-coordinate is \(-\frac{3}{2}\), the reciprocal is \(-\frac{2}{3}\) and half of that is \(-\frac{1}{3}\). So, \(a=-\frac{1}{3}\).

Step 3: Find the equation of the parabola

Substitute the value of a into the standard form of the equation \(x=ay^2\). So the equation will be \(x=-\frac{1}{3}y^2\).

Key Concepts

Parabola with Vertex at Origin

Understanding the structure of a parabola is fundamental to grasping the concept of conic sections in mathematics. A parabola with its vertex at the origin has a very simple and symmetrical shape. The vertex, being at the coordinate point (0,0), makes the analysis of the parabola's properties straightforward.

When dealing with such parabolas, the focus plays a critical role in determining its orientation and width. The focus is a fixed point from which the distance to any point on the parabola is equal to the distance from that point to a line called the directrix. In the case of the given exercise, the parabola opens to the left since the x-coordinate of the focus is a negative value. This negative x-coordinate indicates that the parabola is horizontally oriented, extending to the left of the y-axis.

Focus of a Parabola

The focus of a parabola is a singular point that, along with the directrix, defines the shape of the parabola. Every point on the parabola maintains an equal distance to the focus and the directrix.

In the exercise, the focus is given as \( \left(-\frac{3}{2}, 0\right) \). The negative x-coordinate here is insightful; it informs us that our parabola does not open in the standard rightward direction but instead opens to the left. The position of the focus relative to the vertex also helps us determine the 'stretch' or 'compression' factor of the parabola, known as the value of 'a' in the parabola's equation. The further the focus from the vertex, the wider the parabola will be. By calculating the value of 'a' as done in the solution, the precise 'openness' of the parabola takes shape.

Standard Form of a Parabola

The standard form of a parabola's equation is immensely useful for analyzing its characteristics such as its orientation, vertex, and how it opens. For a parabola that opens left or right, the standard form is \(x = ay^2 + bx + c\), and for one that opens up or down, the standard form is \(y = ax^2 + bx + c\).

In the exercise, we found that the parabola opens to the left, which falls into the first category. With the vertex at the origin and the 'a' value computed from the focus, we can readily visualize and construct this parabola's equation. By substituting the 'a' value into the standard form, the complete equation \(x = -\frac{1}{3}y^2\) describes a leftward-opening parabola compressed horizontally by the factor of \(\frac{1}{3}\). This simple and elegant representation can then be utilized to graph the parabola or analyze its various features and intersections with other geometric entities.