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

Verified step by step guidance

1. Identify the function as a quadratic function in the form \( f(x) = a(x - h)^2 + k \), where \( a = 2 \), \( h = 2 \), and \( k = -4 \).
2. Recognize that the vertex of the parabola is at the point \((h, k) = (2, -4)\). This is the lowest point on the graph since \( a > 0 \), indicating the parabola opens upwards.
3. Determine the axis of symmetry, which is the vertical line \( x = h \). In this case, it is \( x = 2 \).
4. Choose additional points to plot by selecting \( x \)-values around the vertex, such as \( x = 1 \) and \( x = 3 \), and calculate the corresponding \( y \)-values using the function \( f(x) = 2(x - 2)^2 - 4 \).
5. Plot the vertex and the additional points on a coordinate plane, draw the axis of symmetry, and sketch the parabola by connecting the points smoothly, ensuring it opens 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 can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the general shape and properties of parabolas is essential for graphing quadratic functions.

Vertex Form of a Quadratic

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 easy to identify the vertex and the direction in which the parabola opens. In the given function, f(x) = 2(x - 2)² - 4, the vertex is at (2, -4), which is crucial for accurately graphing the function.

Transformations of Functions

Transformations of functions involve shifting, stretching, or reflecting the graph of a function. In the case of the function f(x) = 2(x - 2)² - 4, the '2' indicates a vertical stretch, while '(x - 2)' represents a horizontal shift to the right by 2 units, and '-4' indicates a vertical shift downward by 4 units. Understanding these transformations helps in accurately sketching the graph.