In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = |x|, g(x) = |x| − 2

Verified step by step guidance

1. Step 1: Understand the given functions. The first function, f(x) = |x|, represents the absolute value function, which forms a V-shaped graph with its vertex at the origin (0, 0). The second function, g(x) = |x| - 2, is a transformation of f(x) where the entire graph of f(x) is shifted downward by 2 units.
2. Step 2: Create a table of values for f(x) = |x|. Select integers for x from -2 to 2. For each x value, calculate f(x) by taking the absolute value of x. For example, f(-2) = |-2| = 2, f(-1) = |-1| = 1, f(0) = |0| = 0, and so on.
3. Step 3: Create a table of values for g(x) = |x| - 2. Use the same x values from -2 to 2. For each x value, calculate g(x) by first finding |x| (as in f(x)) and then subtracting 2. For example, g(-2) = |-2| - 2 = 2 - 2 = 0, g(-1) = |-1| - 2 = 1 - 2 = -1, and so on.
4. Step 4: Plot the points for both functions on the same rectangular coordinate system. Use the tables of values to plot the points for f(x) and g(x). Connect the points for f(x) to form the V-shaped graph of the absolute value function. Similarly, connect the points for g(x) to form the graph of the transformed function.
5. Step 5: Analyze the relationship between the graphs of f(x) and g(x). Notice that the graph of g(x) is identical in shape to the graph of f(x), but it is shifted downward by 2 units. This vertical shift is due to the '-2' in the equation g(x) = |x| - 2.

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 that opens upwards, with its vertex at the origin (0,0). Understanding this function is crucial for analyzing transformations applied to it.

Vertical Shifts

Vertical shifts occur when a constant is added to or subtracted from a function. In the case of g(x) = |x| - 2, the graph of f(x) = |x| is shifted downward by 2 units. This transformation affects the y-coordinates of all points on the graph, moving them lower without altering their x-coordinates. Recognizing this concept helps in understanding how the graphs of related functions can be compared.

Graphing Functions

Graphing functions involves plotting points on a coordinate system based on the function's output for various input values. For the functions f(x) = |x| and g(x) = |x| - 2, selecting integer values for x from -2 to 2 allows for a clear visualization of their respective graphs. This process is essential for identifying relationships between functions, such as shifts, reflections, or stretches.