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. Identify the given functions: \(f(x) = |x|\) and \(g(x) = |x| - 2\). Notice that \(g(x)\) is derived from \(f(x)\) by subtracting 2 from the output values.
2. Create a table of values for \(x\) starting from \(-2\) to \$2\(. For each \)x\(, calculate \)f(x) = |x|\( and \)g(x) = |x| - 2\(. For example, when \)x = -2\(, \)f(-2) = |-2| = 2\( and \)g(-2) = 2 - 2 = 0$.
3. Plot the points for both functions on the same coordinate system. For \(f(x)\), plot points like \((-2, 2)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), and \((2, 2)\). For \(g(x)\), plot the corresponding points shifted down by 2 units, such as \((-2, 0)\), \((-1, -1)\), \((0, -2)\), \((1, -1)\), and \((2, 0)\).
4. Draw the graph of \(f(x) = |x|\), which is a 'V' shape with its vertex at the origin \((0,0)\). Then draw the graph of \(g(x) = |x| - 2\), which has the same 'V' shape but shifted vertically downward by 2 units.
5. Describe the relationship: The graph of \(g(x)\) is the graph of \(f(x)\) shifted downward by 2 units. This vertical shift is due to the subtraction of 2 in the function \(g(x) = |x| - 2\).

Key Concepts

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. Its graph is a V-shaped curve with the vertex at the origin (0,0), reflecting all negative inputs as positive outputs. Understanding this shape is essential for graphing and comparing transformations.

Function Transformation - Vertical Shift

A vertical shift occurs when a constant is added or subtracted from a function, moving its graph up or down without changing its shape. For g(x) = |x| - 2, the graph of f(x) = |x| is shifted downward by 2 units, affecting the vertex position and all output values.

Graphing Functions Using Integer Inputs

Selecting integer values for x, such as from -2 to 2, helps in plotting precise points on the coordinate plane. This method simplifies graphing by providing clear reference points to visualize the function's behavior and compare transformations between f and g.