Since my Algebra II experiment years ago, I've been obsessed with students to start making connections between graphs and algebra. Fairly consistently I have students who dismiss graphing and something uninteresting because no one seems to making connections about why it's so interesting. That's it's possible the "x= " they're able to find with a calculator represents something. Pre-Cal has some algebra objectives and I've brought what I learned with that experiment along for the ride.
In addition to emphasizing graphing, I've been demanding a lot of explanations from my kids this year. A go to for a while now has been justifying a situation. Recently I gave kids a graph and three possibilities. In class we focused on the quadratic formula, it's super solving powers, and it's ability to show you the x-values of the intersection points for a pair of functions. In the setup for this question, I didn't specifically ask them to use this method, though most went with that approach. Here's a representative sample:
The brute force method. The student solved all three and made a connection to the picture with their work. Nothing wrong with it.
However, some found other ways:
An excellent observation of graph transformations. The functions used were simple enough that spotting some features was a quick way to accomplish the task.
The next student made a similar find:
Not only did they attempt a quadratic formula solution (apparently done on the calculator and not recorded) but they weren't satisfied that the mere presence of an imaginary number would do the trick. The slope of the line was important!
This last one might be my favorite:
Not only do they grind through the quadratic formula here, but they remember that if those intersection points are solutions, they are useful! I think only one other student (out of ~60) tried this out.
I was pleasantly surprised because I only had one method in mind when designing the question, forgetting that kids could wander in so many interesting directions.