Problem 88 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The graph of the function \(h(x) = -|x+3|\) is a reflection of the absolute value function shifted 3 units to the left and reflected across the x-axis.

Step 1: Graph the absolute value function

First, our base function is \(f(x)=|x|\). This function is a v-shaped graph that has its vertex at the origin (0,0). The lines of this function have a slope of 1 and -1. 'y' increases as 'x' moves away from the origin in both directions.

Step 2: Apply a horizontal shift

Shift the graph of the base function \(f(x)=|x|\) to the left by 3 units to reflect the transformation in \(h(x)=|x+3|\). This shift is due to \(+3\) within the absolute value brackets. Everything to the right of y-axis from the base function now shifts 3 units to the left. The vertex of the graph now shifts from (0,0) to (-3,0).

Step 3: Apply a vertical reflection

The '-' sign in front of the absolute value function results in a reflection across the x-axis. All 'y' values of the shifted graph from step 2 are now negated, flipping the graph upside down. This completes the transformation according to \(h(x) = -|x+3|\).