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|

Verified step by step guidance

1. Start by graphing the parent function f(x) = |x|. This is a V-shaped graph with its vertex at the origin (0, 0). The graph opens upwards, with the left side having a slope of -1 and the right side having a slope of 1.
2. Next, analyze the given function g(x) = |x + 3|. Notice that the expression inside the absolute value, x + 3, indicates a horizontal shift. Specifically, adding 3 inside the absolute value shifts the graph to the left by 3 units.
3. To apply the transformation, take each point on the graph of f(x) = |x| and shift it 3 units to the left. For example, the vertex of f(x) = |x| at (0, 0) will move to (-3, 0).
4. Redraw the graph with the new vertex at (-3, 0). The V-shape remains the same, with the left side having a slope of -1 and the right side having a slope of 1, but the entire graph is now centered at x = -3.
5. Label the graph of g(x) = |x + 3| clearly, and verify that the transformation has been applied correctly by checking a few points. For instance, when x = -3, g(x) = |(-3) + 3| = 0, and when x = -4, g(x) = |(-4) + 3| = 1.

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. In this case, the function g(x) = |x + 3| represents a horizontal shift of the absolute value function f(x) = |x|. Specifically, it shifts the graph 3 units to the left, moving the vertex from (0,0) to (-3,0).

Function Composition

Function composition refers to the process of applying one function to the results of another. In the context of transformations, we can think of g(x) as a composition of the absolute value function with a linear function that shifts the input. Understanding how to manipulate the input of a function is crucial for accurately graphing transformed functions.