Today's soundtrack is Alcest: Le Secret, an absolutely brilliant piece of work. It's a blackgaze (black metal crossed with shoegaze) masterpiece that runs a gamut of emotions and moods: triumphant, chaotic, thoughtful, melodious...
It was re-recorded recently, though I think (as is probably abundantly clear at this point) that the original EP from 2005 stands perfectly well on its own.
This afternoon, I'm working on solving "linear-quadratic equations and quadratic-quadratic equations by graphing" (Pearson's Pre-Calculus 11, p. 386).
When given a graph with multiple systems on it, our solutions will be points where the graphs intersect each other. It's pretty straightforward with a graphing calculator, which is how my textbook asks me to do it. First, we enter the two systems into the [Y=] section, then we graph them. Next, we use [CALC] to determine "5: intersect" at all points where the two systems intersect.
Easy peasy!
If there are two systems on the graph, and at least one of them is quadratic, with the other one either linear or quadratic, there may be either 0, 1, 2, or infinite solutions. We call a solution infinite if both systems are identical, because when that happens, they are in constant overlap. This can only happen if both systems are of the same type - either linear or quadratic.
I did come across a couple of questions where I was asked to look at a graph with multiple systems on it, to write the equations of the systems, and to calculate their solutions. I used my notes on determining the vertex equation of a quadratic function based on its graph and my notes on linear equations to help me with this section.
And with that, this chapter is complete! In the next chapter, I'll be working on solving systems of equations again - but instead of solving them graphically, we'll be solving them algebraically. Muahahaha.