Are there any other methods of plotting absolute value graphs without plugging in arbitrary x values?

Answer 1

Well, we can use information about the equation to determine the shape of the graph. Assume that you have an absolute value equation

y = a|x+b| + c

The b is a horizontal shift (if b is positive, then it is a shift to the left; if it is negative, then it is a shift to the right). The value c is a vertical translation (up if positive, down if negative). Consider a to be the slope of the absolute value function.

If a is positive, then the slope will be a when x>b and the slope will be -b when x
If a is negative (which flips the absolute function upside down), then the slope will be -a when x>b and the slope will be a when x
The pointy part (the minimum or maximum) occurs when x=0, so y=c.

Think about absolute functions as linear functions that sort of bounce off instead of continuing, and it is much easier to interpret their graphs.

Answer 2

The values aren't arbitrary if they're pulled from the original curve. But if they were arbitrary then they still will be now I guess.
I think the way I was taught was probably to find the non absolute value curve and then flip the negative sections upwards.

Answer 3

yes, if you think in terms of transformations of functions. The "parent" function is a V, vertex at the origin, arms of the V with slopes of ±1. y = | x - 3 | will be the same V shifted 3 to the right so the vertex is at (3,0). y = | x + 4 | will be the parent V shifted 4 left. y = |3x| will be the parent V sitting on the origin, but the slopes of the arms will be ±3.

Generally, y = a| x - h | + k will have the vertex at (h,k) and slope of arms ±a.

Answer 4

You're just plugging random points [not just x values] on your graph, to TEST which side of the graph satisfies your equation, and not to plot the AVG itself. You test the graph by plotting test points after you have the graph on the plane.

Hope it helps.