Learning How to Find the Range of a Quadratic Function: f(x) x^2 - 2x - 3

In the realm of algebra, understanding the range of a quadratic function is fundamental. This guide will walk you through how to find the range of the function ( f(x) x^2 - 2x - 3 ) through various methods, including completing the square and utilizing the vertex formula.

1. Completing the Square Method

One of the simplest methods to find the range of the quadratic function ( f(x) x^2 - 2x - 3 ) is by completing the square.

1. **Rewriting the Function:** [ f(x) x^2 - 2x - 3 ] ]

2. **Adding and Subtracting 1 (which is (left(frac{-2}{2}right)^2)):** [ f(x) x^2 - 2x 1 - 1 - 3 (x - 1)^2 - 4 ] ]

3. **Simplification:** [ f(x) (x - 1)^2 - 4 ] ]

Here, since ((x - 1)^2) is a square term, it is always non-negative (i.e., (geq 0)). The lowest value ((x - 1)^2) can take is 0, which occurs when (x 1). Therefore, the function ( f(x) (x - 1)^2 - 4 ) will have its minimum value when ((x - 1)^2 0), which is: [ f(1) 0 - 4 -4 ] ]

This means the minimum value of (f(x)) is (-4), and as (x) moves away from 1 in either direction, ((x - 1)^2) increases, making (f(x)) increase without bound. Therefore, the range of (f(x)) is all real numbers greater than or equal to (-4): [ text{Range: } [-4, infty) ]

2. Using the Vertex Formula

Another method involves using the vertex formula to find the minimum or maximum point of a parabola. For the function ( f(x) ax^2 bx c ), the x-coordinate of the vertex is given by: [ x -frac{b}{2a} ]

1. **Identify the coefficients:** [ a 1, quad b -2, quad c -3 ] ]

2. **Calculate the x-coordinate of the vertex:** [ x -frac{-2}{2 cdot 1} frac{2}{2} 1 ] ]

3. **Substitute (x 1) into the function to find the y-coordinate (minimum value):** [ f(1) 1^2 - 2(1) - 3 1 - 2 - 3 -4 ] ]

Since the parabola opens upwards ((a 1 > 0)), the vertex represents the minimum point. Therefore, the range of (f(x)) is all real numbers greater than or equal to (-4): [ text{Range: } [-4, infty) ]

3. Understanding the Concept of Domain and Range

It is also important to understand the concept of domain and range in function definitions. The domain is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) produced by that function.

A function (f: A rightarrow B) is defined as: 1. A mapping from set (A) to set (B) 2. Set (A) is the domain, and set (B) is the codomain (or range). 3. The set of values actually achieved by the function, known as the image or range, is denoted as (f(A)).

In the function (f(x) x^2 - 2x - 3), the domain is all real numbers, (x in mathbb{R}). The codomain (or range) is ([-4, infty)) based on the minimum value found and the analysis of the parabola’s shape.

Understanding these concepts will greatly help in analyzing the behavior of quadratic functions and their ranges in various applications.