Anybody have any good learning strategies (either tangible or computer simulated) for teaching this to lower-level Algebra 2 students? For example, given the graph of y = x^2, the domain is -Infinity < x < Infinity (or all real numbers) and the range is 0 <= y < Infinity. I went over this last class, and many students' eyes were glazed over with confusion. So I would like to find something they can interact with.

I'd find the largest piece of butcher's paper I can get my hands on, or tape multiple sheets together to get the size I want. I would take this giant sheet of paper to a large open space. I'd stick a sheet of graph paper with the y=x^2 graph and axes drawn on it on the giant sheet of paper. And get students to use a marker to extend the y=x^2 graph and x and y axes out of the sheet of graph paper and onto the butchers paper. When they reach the end of the butchers paper, they can use chalk to extend the graph and axes out onto the ground. That's a really hands on and visual way of showing that the Cartesian plane extends out to infinity and therefore the graph also extends to infinity. And then they can see that the domain goes from - infinity to + infinity and the range never goes below 0.

That's a good idea. The issue they seem to be having is making the transition from discrete domain and range values to a continuous domain and range on an interval, as well as isolating just one of the dimensions at a time and relating that to when they graphed inequalities previously. I might just give them a checklist to go through: What is the smallest x value? What is .... the largest y value? And then have them use those values to write the appropriate compound inequalities.

Use Desmos to graph functions as Desmos has the ability to move along the coordinate plane and zoom in/out. Focus on the end behavior of functions (HSF.IF.B.4).

I remember this was always mind-blowing for my algebra 2 students when I used to teach algebra 2. I always question, too, how relevant this is to lower level students. I found this online, which I tried once, and it went relatively well. It includes both discrete and continuous domains. https://app.box.com/s/cpd2ayhuxblhyaljyy7v I usually use a yardstick and run it across a blown up picture of a graph and move it left/right and up/down and ask them to show me where it starts and stops. Most of them eventually get it, though some may just memorize (i.e. domain of parabola is always all real numbers!)

Yeah, that was my first thought. But I think there are some states, well at least stupid states or deceitful states that would try to make teachers think this was one of the most important things to be taught first.

You could try reminding them that the negative sign is just short hand for the word opposite. So if you have positive numbers, it's like you are facing to your right. In terms of the range, low values represent having a super weak engine for a rocket that you want to launch in the air. The larger values of X are just having a bigger and bigger engine for your rocket. By reminding them the negative values represent opposite, you can say that these just mean you are facing to your left instead of your right because it's the opposite direction. You are still shooting a rocket in the air and still using bigger and bigger engines, but you are just aiming the rocket in an opposite direction.