Problem 84 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=|x+3|$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=|x+3|\) is a 'V' shape that intersects the x-axis at x = -3. It is the graph of the absolute function \(f(x)=|x|\) shifted three units to the left.

Step 1: Graph the Basic Absolute Function

Begin by graphing the absolute value function \(f(x)=|x|\). The graph is a 'V' shape that intersects the origin (0,0). The graph increases equally on both sides from this point.

Step 2: Understand the Transformation

The function \(g(x)=|x+3|\) is a transformation of the basic absolute value function by \(-3\) units horizontally. This means, the graph of the function \(f(x)=|x|\) will shift 3 units to the left (negative direction).

Step 3: Plot the Transformed Function

Next, on the same set of axes, plot the transformed function \(g(x)=|x+3|\) by moving every point on \(f(x)=|x|\) three units to the left. The vertex will now be at the point (-3,0). This will retain its 'V' shape but now intersect the x-axis at x = -3.