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

Verified step by step guidance

1. Start by graphing the basic absolute value function, \( f(x) = |x| \). This graph is a V-shaped graph with its vertex at the origin (0,0) and opens upwards.
2. Identify the transformation needed to graph \( g(x) = |x| + 3 \). This function represents a vertical shift of the graph of \( f(x) = |x| \).
3. To apply the transformation, shift the entire graph of \( f(x) = |x| \) upwards by 3 units. This means every point on the graph of \( f(x) \) will move 3 units higher.
4. The vertex of the new graph \( g(x) = |x| + 3 \) will be at the point (0,3) instead of (0,0).
5. Draw the new graph, ensuring that it maintains the same V-shape as \( f(x) = |x| \), but with the vertex at (0,3) and the arms of the V opening upwards.

Key Concepts

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This function has a V-shaped graph that opens upwards, with its vertex at the origin (0,0). Understanding this function is crucial as it serves as the foundation for graphing transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In this case, adding a constant to the absolute value function, as in g(x) = |x| + 3, results in a vertical shift of the graph upwards by 3 units. This concept is essential for modifying the original graph to represent new functions.

Vertical Shift

A vertical shift occurs when a constant is added to or subtracted from a function's output. For g(x) = |x| + 3, the +3 indicates that every point on the graph of f(x) = |x| is moved up by 3 units. This concept helps in visualizing how the graph changes in relation to the original function.