Problem 89 - Precalculus Essentials

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=-|x+4|+1$$

Step-by-Step Solution

Answer

The function \(g(x)=-|x+4|+1\) is a transformation of the base function \(f(x)=|x|\). It's graph is a V-shaped graph reflected over the x-axis, shifted 4 units to the left and 1 unit upwards with the vertex at (-4,1).

Step 1: Graphing the base function

Plot the function \(f(x)=|x|\). This is a V-shaped graph, where the vertex is at the origin (0,0) and the lines y=x for x>=0 and y=-x for x<0 form the arms of the 'v'.

Step 2: Applying transformations to the base function

The function \(g(x)=-|x+4|+1\) represents a transformation of the base function \(f(x)=|x|\). The '+4' inside the absolute value symbol translates the graph 4 units to the left. The '-' sign in front of the absolute value symbol reflects the graph over the x-axis. Lastly, the '+1' outside the absolute value symbol translates the graph 1 unit upward.

Step 3: Graph the transformed function

Using the transformations interpreted in Step 2, graph the given function \(g(x)=-|x+4|+1\). The vertex of this function will be (-4,1). The left arm of the 'v' (for x<-4) will be the line y=-(x+4)+1. The right arm of the 'v' (for x>=-4) will be the line y=(x+4)+1. The function should be reflected over the x-axis and shifted 4 units to the left and 1 unit upwards compared to the base function graph.