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

Verified step by step guidance

1. Recognize that the given function ƒ(x) = 2(x - 2)² - 4 is a quadratic function in vertex form, where the general form is ƒ(x) = a(x - h)² + k. Here, a = 2, h = 2, and k = -4.
2. Identify the vertex of the parabola, which is the point (h, k). For this function, the vertex is at (2, -4). This is the lowest or highest point depending on the value of a.
3. Determine the direction of the parabola's opening by looking at the coefficient a. Since a = 2 is positive, the parabola opens upwards.
4. Find additional points to plot by choosing x-values around the vertex, substituting them into the function, and calculating the corresponding y-values. For example, calculate ƒ(1), ƒ(3), and ƒ(0) to get points on either side of the vertex.
5. Plot the vertex and the additional points on the coordinate plane, then draw a smooth curve through these points to complete the graph of the parabola.

Key Concepts

Understanding Function Notation and Structure

Function notation, such as ƒ(x), represents a rule that assigns each input x to an output value. Recognizing the structure of the given function, which is a quadratic expression in vertex form, helps in identifying its graph's shape and key features like vertex and axis of symmetry.

Graphing Quadratic Functions in Vertex Form

The vertex form of a quadratic function is ƒ(x) = a(x - h)² + k, where (h, k) is the vertex. This form makes it easy to graph by locating the vertex and determining the parabola's direction and width based on the coefficient a. For example, a positive a opens upward, and a larger |a| value makes the parabola narrower.

Transformations of Functions

Graphing involves understanding transformations such as horizontal shifts (inside the parentheses), vertical shifts (outside the parentheses), and vertical stretching or compressing (multiplying by a). In ƒ(x) = 2(x - 2)² - 4, the graph shifts right by 2 units, down by 4 units, and stretches vertically by a factor of 2.