How do you graph f(x) = ||x|-3|+4?

Answer 1

Since x appears inside absolute value symbols, the graph will look the same for positive and negative values of x, so we can start by focusing on positive values of x. For positive values of x, we have |x-3|, which is zero when x=3, and increases as x moves away from 3. This "V" shape stops at the y-axis (remember, this is for positive values of x), but continues to the right of 3 to infinity. Adding the 4 shifts the graph up by 4 units, so that the tip of the "V" is at the point (3,4). Finally, since the graph on the left side of the y-axis is a mirror image (flipped across the y-axis) of the graph on the right side of the y-axis, the result resembles a "W" with the upward point at (0,7) and the downward points at (-3,4) and (3,4).

Answer 2

For f(x) = | |x| − 3 | + 4, values of x with extremely large absolute values give values of f(x) that are extremely close to x. If x is 1000000 or

-1000000, either number gives a result of f(x) = 1000001, for example. The graph will then slope downward as x comes in from positive and negative infinity until x is either positive or negative 3. This means that the graph will have absolute minimums of f(x) = 4 at x = 3 and x = -3. When x = 0, f(x) = 7 so the graph will have a relative maximum at (0,7). The graph will then appear as a "w" with extremely long outer segments descending downward from (-∞,∞) to (-3,4) and from (∞,∞) to (3,4) and short inner segments going from (-3,4) and (3,4) to (0,7) to complete the graph.