In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = |x+3|

Key Concepts

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This means that for any input x, the function returns x if x is positive or zero, and -x if x is negative. The graph of this function is a V-shape, with its vertex at the origin (0,0), and it is symmetric about the y-axis.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For the function g(x) = |x + 3|, the graph of f(x) = |x| is shifted horizontally to the left by 3 units. Understanding these transformations is crucial for accurately graphing functions derived from basic forms.

Horizontal Shifts

A horizontal shift occurs when a function is modified by adding or subtracting a constant to the input variable. In the case of g(x) = |x + 3|, the '+3' indicates a shift to the left by 3 units. This concept is essential for predicting how the graph of a function will change based on alterations to its equation.