Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. g(x)=(x+2)^2

Verified step by step guidance

1. Identify the basic function: The given function is \( g(x) = (x+2)^2 \), which is a transformation of the basic quadratic function \( f(x) = x^2 \).
2. Determine the transformation: The function \( g(x) = (x+2)^2 \) represents a horizontal shift of the basic quadratic function \( f(x) = x^2 \) to the left by 2 units.
3. Identify the vertex: The vertex of the basic function \( f(x) = x^2 \) is at the origin (0,0). After the horizontal shift, the vertex of \( g(x) = (x+2)^2 \) is at (-2,0).
4. Sketch the graph: Start by plotting the vertex at (-2,0). Since the function is a standard parabola opening upwards, plot additional points symmetrically around the vertex to form the U-shape.
5. Label the graph: Clearly label the vertex and a few other points on the graph to indicate the shape and position of the parabola.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax^2 + 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 functions like g(x) = (x + 2)^2.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on its orientation. For the function g(x) = (x + 2)^2, the vertex can be found by identifying the point where the function reaches its minimum value. In this case, the vertex is at the point (-2, 0), which is crucial for accurately plotting the graph.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. The function g(x) = (x + 2)^2 represents a horizontal shift of the basic quadratic function f(x) = x^2 to the left by 2 units. Understanding these transformations helps in predicting how the graph will change from its parent function.