Summary
Highlights
Introduction to Absolute Value Parent Function00:00:00
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.
Reflections of Absolute Value Functions00:01:00
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.
Domain and Range of Reflected Functions00:01:50
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.
Horizontal Shifts in Absolute Value Functions00:03:06
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.
Vertical Shifts in Absolute Value Functions00:04:34
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].
Combinations of Transformations and Plotting with Tables00:05:55
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).
Graphing with Negative Coefficients and Steeper Slopes00:07:21
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).