Problem 90 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=-|x+4|+2$$

Step-by-Step Solution

Answer

The graph of function \(g(x)=-|x+4|+2\) can be obtained from the graph of the absolute value function \(f(x)=|x|\) by reflecting it over the x-axis, shifting 4 units to the left and then shifting 2 units upwards.

Step 1: Graph the absolute value function

The absolute value function \(f(x)=|x|\) is a V-shaped graph that intersects the origin (0,0), and opens upwards. For any value of \(x\), \(f(x)\) is always non-negative.

Step 2: Understand transformations

In the function \(g(x)=-|x+4|+2\), there are three transformations. The minus sign before the absolute value implies a reflection in the x-axis, the '+4' inside the absolute value function implies a shift 4 units to the left, and the '+2' outside the absolute value function implies a shift 2 units upwards.

Step 3: Apply transformations and graph the function

1. Reflect the graph of the absolute function over the x-axis. 2. Shift the graph 4 units to the left. 3. Shift the graph 2 units upwards. This final graph represents the function \(g(x)=-|x+4|+2\).