Graph each function. See Examples 6–8. g(x) = (x - 4)²

Verified step by step guidance

1. Recognize that the function given is a quadratic function in the form \(g(x) = (x - 4)^2\), which is a transformation of the basic parabola \(y = x^2\).
2. Identify the horizontal shift by noting the expression \((x - 4)\) inside the square. This means the graph of \(y = x^2\) is shifted 4 units to the right.
3. Determine the vertex of the parabola. Since the basic parabola \(y = x^2\) has its vertex at \((0,0)\), the vertex of \(g(x) = (x - 4)^2\) will be at \((4,0)\).
4. Understand the shape and direction of the parabola. Because the coefficient of the squared term is positive (implicitly 1), the parabola opens upwards.
5. To graph the function, plot the vertex at \((4,0)\), then choose values of \(x\) around 4 (such as 3 and 5) to calculate corresponding \(g(x)\) values, and plot these points to sketch the parabola.

Key Concepts

Graphing Quadratic Functions

A quadratic function is a polynomial of degree two, typically written as f(x) = ax² + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the shape and direction of the parabola is essential for graphing functions like g(x) = (x - 4)².

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. For g(x) = (x - 4)², the vertex is at (4, 0). This form makes it easy to identify the vertex and graph the parabola by shifting the basic parabola y = x² horizontally and vertically.

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the graph of a function. In g(x) = (x - 4)², the graph of y = x² is shifted 4 units to the right. Recognizing these transformations helps in quickly sketching the graph without plotting many points.