To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=x^2? What is its domain?

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form ƒ(x) = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which opens upwards if a is positive and downwards if a is negative. For the function ƒ(x) = x², the graph is a parabola that opens upwards with its vertex at the origin (0,0).

Graphing Parabolas

When graphing a quadratic function like ƒ(x) = x², key features include the vertex, axis of symmetry, and direction of opening. The vertex is the lowest point for upward-opening parabolas, and the axis of symmetry is the vertical line that passes through the vertex. The shape of the graph is determined by the coefficient of x², which influences the width and steepness of the parabola.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the quadratic function ƒ(x) = x², the domain is all real numbers, denoted as (-∞, ∞), because you can substitute any real number for x and obtain a valid output. Understanding the domain is crucial for analyzing the behavior of the function and its graph.