Interval Notation for Parabola Calculator

What is Interval Notation?

Interval notation is a concise way to represent sets of real numbers. It uses brackets and parentheses to indicate whether endpoints are included or excluded. Here are the basic symbols:

• [a, b] - All numbers from a to b, including a and b
• (a, b) - All numbers from a to b, excluding a and b
• [a, b) - All numbers from a to b, including a but excluding b
• (a, b] - All numbers from a to b, excluding a but including b
• (a, ∞) - All numbers greater than a
• (-∞, b) - All numbers less than b
• (-∞, ∞) - All real numbers

Interval notation is particularly useful when describing the domain and range of functions, including parabolas.

How to Express Parabolas in Interval Notation

When working with parabolas, interval notation helps define the domain and range of the function. The standard form of a parabola is:

The domain of a parabola is typically all real numbers, which can be expressed as (-∞, ∞). The range depends on the value of 'a':

• If a > 0, the parabola opens upwards and the range is [minimum value, ∞)
• If a < 0, the parabola opens downwards and the range is (-∞, maximum value]

To find the minimum or maximum value, you can use the vertex form of the parabola:

Where (h, k) is the vertex of the parabola. The minimum or maximum value is k.

Examples of Interval Notation for Parabolas

Let's look at a few examples to see how interval notation applies to parabolas.

Example 1: y = x² - 4x + 3

First, let's find the vertex of the parabola. We can complete the square:

The vertex is at (2, -1). Since the coefficient of x² is positive, the parabola opens upwards. The minimum value is -1. Therefore:

• Domain: (-∞, ∞)
• Range: [-1, ∞)

Example 2: y = -2x² + 8x - 5

Again, let's find the vertex by completing the square:

The vertex is at (2, 3). Since the coefficient of x² is negative, the parabola opens downwards. The maximum value is 3. Therefore:

• Domain: (-∞, ∞)
• Range: (-∞, 3]