Summary
Highlights
The absolute value parent function, y = |x|, graphs as a V-shape opening upwards. To plot it, choose a center point (0,0), then select two points to the right and left, calculating their corresponding y-values. For example, |0|=0, |1|=1, |-1|=1, |2|=2, |-2|=2.
If a negative sign is outside the absolute value function (e.g., y = -|x|), the graph reflects over the x-axis, opening downwards. If the negative sign is inside (e.g., y = |-x|), it does not change the graph, as the absolute value of any negative number is positive, so the graph still opens upwards.
The domain for all standard absolute value functions without fractions, radicals, or logarithms is all real numbers (negative infinity to positive infinity). For y = -|x|, the range is from negative infinity to 0 (inclusive). For y = |x|, the range is from 0 (inclusive) to positive infinity.
An absolute value function like y = |x + 2| shifts two units to the left, while y = |x - 3| shifts three units to the right. These graphs still open upwards with a slope of 1 (or -1 on the left side) from the vertex.
Functions like y = |x| + 2 shift up two units, and y = -|x| - 3 shift down three units. Vertical shifts affect the range. For y = |x| + 2, the range is [2, infinity). For y = -|x| - 3, the range is (-infinity, -3].
When combining transformations, such as y = |x - 2| + 3, the graph shifts right two units and up three units, with the vertex at (2,3). To plot points, set the inside of the absolute value to zero to find the x-coordinate of the vertex. Then choose points around this x-value to calculate y-values. The range for this example is [3, infinity).
For functions like y = 4 - |x + 1|, the graph shifts left one unit, up four units, and opens downwards due to the negative sign, with the vertex at (-1,4). The slope is 1 (or -1) but reflected. For y = 2|x - 1| + 3, the slope is 2, making the graph steeper, shifted right one and up three. For y = 5 - 3|x - 1|, the graph shifts right one, up five, and opens downwards with a slope of 3 (steeper and reflected).