Problem 2 - Precalculus: Functions and Graphs

Question

Exer. \(1-12\) : Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$x^{2}=-3 y$$

Step-by-Step Solution

Answer

Vertex: \((0,0)\); Focus: \((0, -\frac{3}{4})\); Directrix: \(y = \frac{3}{4}\).

Step 1: Identify the form of the equation

The given equation is \(x^2 = -3y\), which is similar to the standard form \(x^2 = 4py\). Here, we can see that \(4p = -3\).

Step 2: Determine the values of \(p\)

From the equation \(4p = -3\), we solve for \(p\) by dividing both sides by 4. Therefore, \(p = -\frac{3}{4}\).

Step 3: Calculate the vertex

For the equation \(x^2 = 4py\), the vertex is at the origin, \((0,0)\).

Step 4: Find the focus

The focus of the parabola \(x^2 = 4py\) is located at \((0, p)\). Substituting the value of \(p = -\frac{3}{4}\), the focus is at: \((0, -\frac{3}{4})\).

Step 5: Determine the directrix

The directrix of the parabola \(x^2 = 4py\) is the line \(y = -p\). Therefore, substituting \(p = -\frac{3}{4}\), the directrix is \(y = \frac{3}{4}\).

Step 6: Sketch the graph

Plot the vertex at \((0,0)\), the focus at \((0, -\frac{3}{4})\), and draw the directrix line \(y = \frac{3}{4}\). Sketch the parabola opening downward from the vertex, as the equation \(x^2 = -3y\) indicates it opens in the negative y-direction.

Key Concepts

Vertex

In the realm of parabolas, the vertex is a crucial point. It's essentially the "tip" of the parabola where it changes direction. For every parabola in standard form like \[ x^2 = 4py \], the vertex takes center stage at \((0, 0)\).
To better understand:

• Graphic Representation: Imagine placing a pin at the vertex, allowing the parabola to swing open around this point. It’s literally the hinge point of the parabola.
• Mathematical Significance: The vertex is often used in determining other components of the parabola, such as its focus and directrix.

In practical scenarios, knowing the vertex helps to simplify the process of graphing, as it forms the foundation from which other elements are determined.

Focus

The focus of a parabola is another fascinating aspect. Think of it as a secret treasure spot: a fixed point that influences the shape of the parabola. When the path traced by points that all "look" towards this spot is plotted, it creates the arching structure of a parabola.
For the standard form \( x^2 = 4py \), the focus is positioned at \((0, p)\). In our specific equation \( x^2 = -3y \), we've found:

• Value of \(p\): Calculated to be \(-\frac{3}{4}\).
• Focus Point: Substituting \( p \) gives us the focus at \((0, -\frac{3}{4})\).

The importance of the focus:

• Each point on the parabola is equidistant from the focus and the directrix. This unique property helps define a parabola geometrically.
• Understanding the focus aids in the construction and interpretation of parabolic shapes, such as satellite dishes, where signals converge at this point.

Directrix

Another pivotal concept is the directrix. This term refers to a fixed line related to the parabola, acting as a boundary that a parabolic curve will mirror itself against. It's akin to the rule that guides and balances the curve.
In an equation like \( x^2 = 4py \), the directrix is defined as \( y = -p \). Our specific example, where \( p = -\frac{3}{4} \), results in:

• Equation of the Directrix: \( y = \frac{3}{4} \).

Why does the directrix matter?

• Its consistent position helps to determine the width and orientation of the parabola, complementing the focus by staying equidistant from points across the curve.
• Recognizing the relationship between the focus and directrix is fundamental in applications like robotics or when dealing with reflective surfaces, where precision is key.

The interplay between the focus and directrix truly crafts the parabola’s defining traits, showcasing the intricate balance of geometry.