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) = (1/2)(x − 1)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 for \( g(x) = \frac{1}{2}(x - 1)^2 \). Notice that the function involves a horizontal shift, a vertical stretch/compression, and a vertical shift.
3. Recognize the horizontal shift: \( (x - 1)^2 \) indicates a shift to the right by 1 unit. This moves the vertex of the parabola from (0,0) to (1,0).
4. Identify the vertical compression: The factor \( \frac{1}{2} \) in front of \( (x - 1)^2 \) compresses the parabola vertically by a factor of \( \frac{1}{2} \), making it wider.
5. Graph the transformed function \( g(x) = \frac{1}{2}(x - 1)^2 \) by applying the horizontal shift and vertical compression to the original parabola \( f(x) = x^2 \). The vertex is now at (1,0), and the parabola is wider than the standard parabola.

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 applying transformations to graph other quadratic functions.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For quadratic functions, common transformations include vertical and horizontal shifts, which can be represented by modifying the function's equation. For example, in g(x) = (1/2)(x - 1)², the graph is shifted right by 1 unit and vertically compressed by a factor of 1/2, which alters its appearance while maintaining its parabolic shape.

Vertex Form of Quadratic Functions

The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it easier to identify the vertex and understand how transformations affect the graph. In the function g(x) = (1/2)(x - 1)², the vertex is at (1, 0), indicating the point where the parabola reaches its minimum value, which is crucial for accurately graphing the transformed function.