Problem 83 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The function \(g(x) = |x + 4|\) is the same as the base function \(f(x) = |x|\) but shifted 4 units to the left.

Step 1: Graph the base function \(f(x) = |x|\)

To graph \(f(x) = |x|\), remember that the graph of the absolute value function forms a 'V' shape. The lowest point of the 'V' is at the origin, (0,0). It increases to the right and left, showing all output values (y-values) are non-negative.

Step 2: Understand the transformation

The function \(g(x) = |x + 4|\) represents a horizontal shift of the base function. Since it's \(+4\) inside the absolute value, the graph will shift 4 units to the left. This is because the value inside the absolute function will hit zero, and thus the minimum point of the 'V', 4 units earlier.

Step 3: Graph the new function \(g(x) = |x + 4|\)

To graph \(g(x) = |x + 4|\), draw the same 'V' shape as the base function, but shift the minimum point 4 units to the left to (-4,0). The graph will increase from this minimum point, making all y-values from this point onwards positive.