[jyl]

#2 is easy, its just the terminology that is unfamiliar.

The "root" of y = f(x) means x for which y = 0.
E.g. y = (x+1) has root x = -1. That's one root.
To add another root, just add another term in the form (x+b)
E.g. y = (x+1)*(x-1). The second root is x = 1.
And so on. So, you end up w/ a polynomial of the form
y = (x+a)*(x+b)* . . . *(x+e) where a . . . e are constants of your choice.
And this polynomial has five roots. x = -a, -b, . . . -e. At each of these roots, y = 0, meaning the curve intersects the x-axis.
"Vertex", in this context, means local maxima or local minima.
As you move from one root to the adjacent root, the curve rises (or falls) from the x-axis, then falls (or rises) to intersect the x-axis again. So there is (at least) one vertex between each pair of adjacent roots. Five roots implies 4 vertices.

#3 as well.

The kids have learned only a few function types so far. The main ones are linear ones like y = a*x +b (which describe a straight line that may be sloped) and quadratic ones like y = a*x^2 + b*x + c (that describe a parabolic curve).
So, you can see that a sloping line and a parabola could easily intersect at two and only two points.
I told her to draw such a picture, then figure out the equations that would describe the curves, then solve for the intersection points.
For example, y = 2x +10 and y = x^2 will intersect at two points. To solve, she'll probably have to do more quadratic equation stuff.

I think her teacher is pretty good. The problems she sets for the kids are well thought out and quite instructive. Sometimes the kids don't fully grasp the material, but I don't think they need perfect comprehension at this point.

Next year, I would assume they move on to trig. Not sure what else - can't spend a whole school year on trigometry.

It occurs to me that, at this rate, she will likely graduate from high school (we have it picked out, will be a public one) with far more education than I had at that point. I graduated HS with some math but very little physics, zero chemistry or biology, self taught in English and history, and basically not much else. In fact, my HS education basically sucked, and it was supposed to be a good HS (El Segundo in L.A.).

We moan about education today, but there is good education to be had out there if the kids and parents make it a priority.