In mathematics, a parabola is a U-shaped curve that can be defined in several ways. One common definition involves a fixed point called the focus and a fixed line called the directrix. A parabola is the set of all points that are equidistant from the focus and the directrix. Understanding the properties of parabolas, such as their vertex and focus, is crucial in various fields, including physics, engineering, and computer graphics. This guide will provide a step-by-step approach to finding the vertex and focus of a parabola, using the example equation y² - 4y + 12x - 8 = 0. We will delve into the process of converting the given equation into the standard form, identifying the key parameters, and ultimately determining the vertex and focus coordinates.
Understanding the Standard Forms of a Parabola
Before we dive into solving the problem, it's essential to understand the standard forms of a parabola. There are two main standard forms, depending on whether the parabola opens horizontally or vertically.
Horizontal Parabolas
A horizontal parabola opens either to the left or to the right. The standard form equation for a horizontal parabola is:
(y - k)² = 4p(x - h)
Where:
- (h, k) represents the vertex of the parabola.
- p is the distance from the vertex to the focus and from the vertex to the directrix.
- If p > 0, the parabola opens to the right.
- If p < 0, the parabola opens to the left.
The focus of a horizontal parabola is located at the point (h + p, k). The directrix is a vertical line with the equation x = h - p.
Vertical Parabolas
A vertical parabola opens either upwards or downwards. The standard form equation for a vertical parabola is:
(x - h)² = 4p(y - k)
Where:
- (h, k) represents the vertex of the parabola.
- p is the distance from the vertex to the focus and from the vertex to the directrix.
- If p > 0, the parabola opens upwards.
- If p < 0, the parabola opens downwards.
The focus of a vertical parabola is located at the point (h, k + p). The directrix is a horizontal line with the equation y = k - p.
Step-by-Step Solution for Finding the Vertex and Focus
Now, let's apply this knowledge to the given equation: y² - 4y + 12x - 8 = 0. Our goal is to rewrite this equation in one of the standard forms mentioned above.
Step 1: Rearrange and Complete the Square
The first step is to rearrange the equation, grouping the y terms together and moving the x term and the constant to the other side:
y² - 4y = -12x + 8
Next, we complete the square for the y terms. To do this, we take half of the coefficient of the y term (-4), square it ((-2)² = 4), and add it to both sides of the equation:
y² - 4y + 4 = -12x + 8 + 4
This simplifies to:
y² - 4y + 4 = -12x + 12
Step 2: Factor and Rewrite in Standard Form
Now, we factor the left side as a perfect square and factor out a constant from the right side:
(y - 2)² = -12(x - 1)
This equation is now in the standard form of a horizontal parabola: (y - k)² = 4p(x - h).
Step 3: Identify the Vertex
By comparing our equation (y - 2)² = -12(x - 1) with the standard form (y - k)² = 4p(x - h), we can identify the vertex (h, k). In this case:
- h = 1
- k = 2
Therefore, the vertex of the parabola is (1, 2). This is a crucial point, as it serves as the center of our parabola and a reference for locating the focus and directrix.
Step 4: Determine the Value of p
Next, we need to find the value of p. We know that 4p is the coefficient of the (x - h) term, which is -12 in our equation:
4p = -12
Dividing both sides by 4, we get:
p = -3
The value of p is negative, which indicates that the parabola opens to the left.
Step 5: Calculate the Focus
For a horizontal parabola, the focus is located at the point (h + p, k). We know that h = 1, k = 2, and p = -3. Plugging these values into the formula, we get:
Focus = (1 + (-3), 2) = (-2, 2)
Therefore, the focus of the parabola is (-2, 2). The focus is a key point inside the curve of the parabola, and it plays a crucial role in the parabola's reflective properties.
Summarizing the Results
In summary, for the parabola given by the equation y² - 4y + 12x - 8 = 0, we have found:
- Vertex: (1, 2)
- Focus: (-2, 2)
Additional Insights and Applications
Understanding how to find the vertex and focus of a parabola is not just an academic exercise. It has practical applications in various fields:
- Optics: Parabolic mirrors are used in telescopes and satellite dishes because they can focus parallel rays of light (or radio waves) to a single point, the focus.
- Antennas: The shape of a parabolic antenna helps to concentrate signals at the focus, improving signal strength.
- Architecture: Parabolic arches are used in bridges and buildings for their structural strength and aesthetic appeal.
- Projectile Motion: The path of a projectile, such as a ball thrown in the air, often follows a parabolic trajectory (ignoring air resistance).
By mastering the techniques for analyzing parabolas, you gain valuable tools for understanding and solving problems in these areas.
Conclusion
This guide has provided a comprehensive explanation of how to find the vertex and focus of a parabola. By completing the square, rewriting the equation in standard form, and applying the appropriate formulas, we successfully determined the vertex and focus for the example equation y² - 4y + 12x - 8 = 0. Remember, the vertex is the turning point of the parabola, and the focus is a crucial point inside the curve that defines its reflective properties. With practice, you can confidently analyze and understand parabolas in various mathematical and real-world contexts. The ability to identify these key features of a parabola is fundamental to understanding its properties and applications, making this a valuable skill in mathematics and related fields.
