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

Verified step by step guidance

1. Start by graphing the standard quadratic function \( f(x) = x^2 \). This is a parabola that opens upwards with its vertex at the origin (0,0).
2. Identify the transformation needed to graph \( g(x) = (x - 2)^2 \). This function is a horizontal shift of the standard parabola.
3. Recognize that the expression \( (x - 2) \) indicates a shift to the right by 2 units. This is because the transformation \( x - h \) shifts the graph to the right by \( h \) units.
4. Apply the horizontal shift to the graph of \( f(x) = x^2 \). Move every point on the parabola 2 units to the right.
5. Plot the new graph of \( g(x) = (x - 2)^2 \). The vertex of the parabola is now at (2,0), and the shape of the parabola remains unchanged, still opening upwards.

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.