Problem 81 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=|x|+4$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=|x|+4\) is the same as the V-shape of an absolute value function \(f(x)=|x|\), but its entire graph shifts upwards by 4 units.

Step 1: Graph the absolute value function \(f(x) = |x|\)

Begin by creating a graph for \(f(x)= |x|\). This is a V-shaped graph that intersects the origin (0,0). The line is linear with a 45 degree slope, moving to the right for positive x values and to the left for negative x values.

Step 2: Understand the concept of transformation

In the function \(g(x) = |x| + 4\), the '+4' outside the absolute value symbol represents a translation or vertical shift of the function. More specifically, it moves the entire graph of the function 4 units up, as '+' indicates upwards movement on the y-axis.

Step 3: Graph the transformed function \(g(x) = |x| + 4\)

Translate the absolute value graph upward by 4 units, without changing its shape. The vertex point will shift from the origin (0,0) to the new position (0,4). The function will retain its V-shape, but now the lowest point is on the y-axis at y = 4.