Problem 22 - Precalculus with Limits

Question

In Exercises 19-32, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: \((-\frac{3}{2}, 0)\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola with focus at \(-\frac{3}{2}, 0\) and vertex at the origin is: \(x = \frac{3}{2}y^2\).

Step 1: Identify the Key Components

Here, the given focus is \((-\frac{3}{2}, 0)\). This implies that the parabola opens to the left as the x-coordinate of the focus is negative.

Step 2: Find 'a' Value Using Focus

The distance between the vertex and the focus is the value of 'a'. From the vertex at the origin to the focus at \(-\frac{3}{2}\), 'a' equals \(\frac{3}{2}\).

Step 3: Writing the Equation

Given that the parabola opens to the left, the standard equation is in the form \(x = a \cdot y^2\). Plugging 'a' into the formula, we get: \(x = \frac{3}{2}y^2\).

Key Concepts

Focus of a Parabola

The focus of a parabola is a critical point that lies inside the parabola. It's important because it helps determine the direction in which the parabola opens. For every point on the parabola, the distance to the focus is equivalent to the distance to the directrix, a fixed line parallel to the axis of symmetry.
When dealing with parabolas, the focus works together with the vertex and serves as a cornerstone for plotting the graph. In this exercise, the focus is given as \(( -\frac{3}{2}, 0)\).

• The x-coordinate tells us how far left or right from the origin the focus is placed.
• The y-coordinate indicates height above or below the origin but is zero in this case, meaning the focus is along the x-axis.

Knowing the focus helps in determining which direction the parabola opens. Since the x-coordinate of the focus here is negative, it points to the left side, indicating that the parabola opens leftward.

Vertex of a Parabola

The vertex of a parabola is its peak point, often regarded as its turning point. For parabolas that open left, right, upwards, or downwards, the vertex can either be the highest or the lowest point. It's vital in defining the shape and position of the parabola on a graph.
In the given problem, the vertex is located at the origin, \((0, 0)\). This makes it easier to find other characteristics, as the calculations are simplified without additional transformations.

• The vertex provides a reference point or starting point for graphing.
• With the vertex at the origin, any movement along x or y is directly attributed to other properties of the parabola, like the focus or a value.

The position of the vertex gives the "central point" from which the shape of the parabola is symmetric, ensuring that the distance and angles are consistent on both sides of the parabola.

Standard Form of a Parabola

The standard form of a parabola's equation provides a clean and easy-to-use tool for displaying the parabola on a graph. This form looks different depending on whether the parabola opens up, down, left, or right.
For parabolas with a vertical axis, the standard form is \(y = ax^2 + bx + c\). But for horizontal parabolas, like the one in the exercise, the standard form is \(x = ay^2 + bx + c\). This reflects how the parabola opens sideways.

• The 'a' value in the equation indicates the "stretch" of the parabola. It shows how wide or narrow the parabola is.
• When 'a' is positive, the parabola opens to the right, and when 'a' is negative, it opens to the left.

In the exercise given, because the parabola opens to the left, it's described by \(x = \frac{3}{2}y^2\). This means the variable part \(ay^2\) dictates the movement in the x-direction, effectively capturing the interaction between the y and x movements.