Domain And Range Of A Parabola X² - 4x + 4y - 16 = 0

Jul 31, 2025 by ADMIN 53 views

Unveiling the Domain and Range of a Parabola: A Comprehensive Guide

Hey guys! Let's dive into the fascinating world of parabolas and explore how to determine their domain and range. We'll be using the equation x² - 4x + 4y - 16 = 0 as our guide. So, buckle up and get ready to unravel the secrets of these captivating curves!

Understanding the Parabola Equation

To kick things off, let's first understand the equation that defines our parabola: x² - 4x + 4y - 16 = 0. This equation is in a general form, and to make it easier to analyze, we need to transform it into a more familiar form called the vertex form. The vertex form of a parabola equation helps us quickly identify key features like the vertex (the turning point of the parabola) and the direction in which the parabola opens.

Transforming to Vertex Form

The process of converting the general form to vertex form involves a technique called completing the square. This technique allows us to rewrite the equation in a way that reveals the vertex coordinates directly. Here's how it works:

Isolate the y-term: First, let's isolate the term containing 'y' on one side of the equation. We can do this by adding 16 and subtracting x² and -4x from both sides:
```
4y = -x² + 4x + 16
```
Divide by the coefficient of y: To get 'y' by itself, divide both sides of the equation by 4:
```
y = (-1/4)x² + x + 4
```
Complete the square for the x-terms: Now, let's focus on the x-terms: (-1/4)x² + x. To complete the square, we need to add and subtract a constant term that will allow us to rewrite these terms as a perfect square. The constant we need is (b/2a)², where 'a' is the coefficient of x² and 'b' is the coefficient of x. In our case, a = -1/4 and b = 1. So, (b/2a)² = (1 / (2 * -1/4))² = (-2)² = 4.

Let's add and subtract this constant inside the equation:
```
y = (-1/4)(x² - 4x) + 4
```
Now complete the square inside the parenthesis:
```
 y = (-1/4)(x² - 4x + 4 - 4) + 4
```
Rewrite as a squared term:
```
y = (-1/4)(x - 2)² + 1 + 4
```
Simplify: Finally, let's simplify the equation:
```
y = (-1/4)(x - 2)² + 5
```

Now we have the equation in vertex form: y = (-1/4)(x - 2)² + 5. This form tells us a lot about the parabola.

Decoding the Vertex Form

The vertex form of a parabola equation is generally written as:

y = a(x - h)² + k

where:

(h, k) represents the vertex of the parabola.
a determines the direction and width of the parabola.

Comparing this to our equation, y = (-1/4)(x - 2)² + 5, we can identify the following:

Vertex: (h, k) = (2, 5)
a: a = -1/4

Interpreting the Values

The vertex (2, 5) tells us the turning point of the parabola. It's the highest point on the parabola since 'a' is negative.
The value of 'a' (-1/4) tells us that the parabola opens downwards (because 'a' is negative) and is wider than the standard parabola y = x² (because the absolute value of 'a' is less than 1).

Determining the Domain and Range

Now that we have a good understanding of our parabola, let's determine its domain and range. The domain refers to all possible x-values that the parabola can take, while the range refers to all possible y-values.

Domain of the Parabola

For parabolas, the domain is always all real numbers. This means that you can plug in any x-value into the equation, and you'll get a corresponding y-value. In mathematical notation, we write this as:

Domain: (-∞, ∞)

This indicates that the parabola extends infinitely in both the positive and negative x-directions.

Range of the Parabola

The range is a bit more interesting because it depends on the vertex and the direction the parabola opens. Since our parabola opens downwards and has a vertex at (2, 5), the highest y-value it can reach is 5. It then extends downwards infinitely. Therefore, the range is all y-values less than or equal to 5. In mathematical notation, we write this as:

Range: (-∞, 5]

The square bracket indicates that 5 is included in the range.

Final Answer

So, to answer the original question, the domain and range of the parabola modeled by the equation x² - 4x + 4y - 16 = 0 are:

Domain: (-∞, ∞)
Range: (-∞, 5]

Therefore, the correct answer from the options provided is A. D: (-∞, 5]: R: (-∞, ∞). Oops! It seems there's a slight mix-up in the options. The correct representation should be D: (-∞, ∞) and R: (-∞, 5].

Graphing the Parabola

To visualize this, let's quickly discuss how to graph the parabola. We already know the vertex (2, 5) and the direction it opens (downwards). To get a better sense of the shape, we can find a few additional points by plugging in some x-values into the equation. For example:

When x = 0:
```
y = (-1/4)(0 - 2)² + 5 = (-1/4)(4) + 5 = 4
```
So, the point (0, 4) is on the parabola.
When x = 4:
```
y = (-1/4)(4 - 2)² + 5 = (-1/4)(4) + 5 = 4
```
So, the point (4, 4) is also on the parabola.

Plotting these points along with the vertex will give you a clear picture of the parabola's shape and confirm our domain and range calculations.

Key Takeaways

Vertex Form is Key: Converting the general form of a parabola equation to vertex form (y = a(x - h)² + k) makes it easy to identify the vertex (h, k) and the direction of opening (determined by 'a').
Domain is Always Real Numbers: For parabolas, the domain is always (-∞, ∞).
Range Depends on Vertex and Direction: The range depends on whether the parabola opens upwards or downwards and the y-coordinate of the vertex.
Completing the Square: This technique is crucial for transforming the general form to vertex form.

Wrapping Up

I hope this comprehensive guide has helped you understand how to determine the domain and range of a parabola. Remember, understanding the vertex form and the significance of the 'a' value are crucial for this process. Keep practicing, and you'll become a parabola pro in no time! If you have any questions, feel free to ask. Happy graphing, guys!