Help me out here, Algebra II people, we have a problem.
The answer I'm looking for, is "where does the given parabola intersect the line y = 7?"
One of the first things I decided to cover in Pre-Cal this year was solving quadratic equations. It creeps up on you in Calculus and beyond that I don't want my students to wait until they're 22 to realize its value like I did.
When I introduced the topic, I asked "why do quadratics have two solutions?" In every class someone suggested it had something to do with the x-axis. I pushed and wondered "what's your obsession with the x-axis?"
I tried to confront this problem last year with my method for introducing the quadratic formula. Students saw the quadratic formula only as a means for finding x-intercepts. They are not totally wrong, but they don't know the truth.
Part of this is the fault of textbooks. As I noticed last year, there is an obsession with giving students quadratics that are equal to 0 and equating roots, solutions, and zeros as like terms. When really, what they need to understand, is you can construct a quadratic which in turn has x-intercepts that are equivalent to the intersection (or lack thereof) of the starting expressions.
Compounding the problem is then showing them half a dozen methods for solving, muddying the waters the whole way. Should I factor? Should I complete the square? What about this formula thing? Students have no idea where to start. Every class admitted this to me. All of them could tease out factoring and the quadratic formula as methods, but very few could describe when one is more desirable.
I'd love to admit that my returning Pivot students nailed this question like they should have, but that wasn't true.
The traditional treatment of quadratics and what the two solutions represent does more harm than you think.