Problem 86 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$h(x)=|x+3|-2$$

Step-by-Step Solution

Answer

The transformed graph of the function \(h(x)=|x+3|-2\) is a 'V' shape that is shifted 3 units to the left and 2 units down from the original graph of the absolute value function.

Step 1: Graph the Absolute Value Function

Start by graphing the function \(f(x)=|x|\). This function forms a 'V' shape, with the vertex at the origin (0,0). The left side of the graph is a reflection of the right side over the y-axis.

Step 2: Identify the Transformations

The function \(h(x)=|x+3|-2\) includes two transformations of the absolute value function. The '+3' inside the absolute value brackets shifts the graph 3 units to the left, and the '-2' outside the brackets shifts the graph 2 units downwards.

Step 3: Apply the Transformations

Apply the transformations identified in the previous step to the original graph of \(f(x)=|x|\). Shift the graph 3 units to the left by subtracting 3 from every x-coordinate, then shift it 2 units down by subtracting 2 from every y-coordinate.

Step 4: Graph the Transformed Function

Lastly, sketch the graph of the transformed function \(h(x)=|x+3|-2\) with the new coordinates. It should still look like a 'V' shape, but it's now repositioned according to the transformations.