In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (x − 2)²

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The graph of a quadratic function is a parabola, which opens upwards if 'a' is positive and downwards if 'a' is negative. Understanding the basic shape and properties of the standard quadratic function f(x) = x² is essential for analyzing transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For the function g(x) = (x - 2)², the transformation is a horizontal shift to the right by 2 units. Recognizing how these transformations affect the graph of the parent function allows for accurate graphing of modified functions.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on its orientation. For the standard quadratic function f(x) = x², the vertex is at the origin (0,0). In the transformed function g(x) = (x - 2)², the vertex shifts to (2,0), which is crucial for accurately plotting the graph and understanding its properties.