Problem 85 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The graph of the function \(h(x) = |x+4|-2\) is the graph of the absolute value function \(f(x)=|x|\), shifted to the left by 4 units and down by 2 units. The vertex of the graph is at the point (-4,-2).

Step 1: Draw Base Function

Start by sketching the base function \(f(x)=|x|\) which is a V-shape centered at the origin. This means the vertex of the V is at (0,0), and the left and right arms of the v shape are lines that have slope of -1 and 1 respectively.

Step 2: Apply horizontal shift

The expression \(x+4\) inside the absolute value brackets instructs a horizontal shift. The graph of the function will move to the left 4 units because of '+4'. Shift the vertex from (0,0) to (-4, 0) and the rest of the graph accordingly.

Step 3: Apply vertical shift

The number \(-2\) outside the absolute value brackets suggests a vertical shift. This will move the graph downward by 2 units. Shift the vertex from (-4,0) to (-4,-2) and the rest of the graph accordingly.

Step 4: Draw the Final Graph

After applying the transformations correctly, sketch the final graph. The graph should look like the original V-shaped graph but shifted 4 units to the left and 2 units down.