Do students learn implicit equations (such as $x^2+y^2-r^2 = 0$) and parametric equations (e.g., $x=a t^2,\;y= 2 a t$) in a first course in algebra, which in the US would be early high school, maybe 9th grade? Or not until pre-calculus or calculus, late high school or early college? Perhaps parametric equations are not part of standard curricula?
I'm trying to gauge how much it is reasonable for me to assume when writing for high-school students.
I am in a US high school. The implicit equations look like "conics", and are part of the junior (3rd year) class typically called Trigonometry with Algebra. Parametrics are part of the course labeled pre-calculus, which comprises either juniors who will take calculus as seniors, or as seniors in preparation of college calculus.
(Although the OP asks for USA, this answer is for maths in England and Wales, for context and comparison)
In Britain, Higher tier GCSE students (age 14-16) will meet $ax+by=c$ as the equation of a line (often in the context of simultaneous equations) and $x^2 +y^2 = r^2$ as the equation of a circle, and perhaps $xy=k$ as representing an inverse proportionality between $x$ and $y$. Moreover they should have the skills to rearrange a formula to change the subject, so they should be able to change $ax+by=c$ to $y=mx+c$, to determine the gradient of a line.
Foundation tier GCSE students may be taught to plot $ax+by=c$ in simple cases, eg $x+y=6$
However implicit and parametric curves are explicitly taught for the first time in the second year of A-level (typically the last year of high school age 17-18) and so only to those who have opted to study maths beyond age 16. They are taught to do implicit differentiation, to find the cartesian form of a parametric curve (including by the use of trigonometric identities), to find an expression for $\frac{dy}{dx}$ in terms of the parameter, and to find the area under a parametric curve.
So the majority of high school students in the UK will never be taught about implicit and parametric curves.
The community college where I teach puts an introduction (lines, circles, ellipses, parabolas) in precalculus, and then covers it again in vector calculus. So, this would be first year and again in second year of undergrad.
I think college prep kids (your general population at a "Sister" school) will have seen this stuff. But it will not be as ingrained as polynomial differentiation or the quadratic equation. So you should somewhat understand the bottom half of your classes, having a slight hiccup, needing to actually use these concepts.
Circle equation:
That's classic "algebra 2", as part of the "conics". They will learn the equation and graph it on Cartesian coordinates. Not the 9th grade first algebra class, but a standard part of the 11th grade. (10th for kids who are accelerated a year and heading to AP Calc).
So, yes absolutely standard high school topic. Maybe not as strongly learned as polynomial graphs. There is a sort of veneration and cleaving to "functions" as opposed to "relations" as if God loved the former more. And they are simpler, with the whole no more than one y per x. And functions are probably more important in technical subjects like chemistry or physics. So the circle (or ellipse) equation can be a bit of a sidelight. But I'd argue even here, it's got some importance as we think about centripetal forces or orbits or the like. In any case, yeah "they had it", even if it (and arguably a lot of stuff you teach) is sort of off the classic, functional, cannon-ball-hitting-things, algebra to calculus track.
In precalculus (nominally 12th grade, 11th for "GT crowd"), the circle equation probably gets reinforced slightly. Still remember learning the function concept and the relation concept (relations get no love here, look at the arctangent debates). But the circle is a classic example of a relation, so it was getting double duty as an example (in Cartesian coordinates). I.e. getting some reinforcement.
Parametric equations:
I had this in pre-calculus (second semester, "analytic geometry"). Nominally 12th grade class, but all the "GT crowd" kids had it in 11th, setting up for AP. Decent public high school, but not top of the heap. 1980s. And if you look in books, you will still see it. I remember graphing cloverleafs or the like, on r-theta graph paper. (Loved that paper...love all cool graph papers, like semilog or maneuvering boards or triangular phase diagrams. Actually...now that I think about it, if your kids are weak, I would definitely hand out some of that graph paper and have them do stuff by hand, at least a little bit...it can be powerful learning to hand graph as opposed to only relying on the computer magic black box.) So r theta was definitely covered.
I could see how weaker kids might not have it until college, and even in the 80s, there was this trend of calculus textbooks to say "with analytic geometry". So, there's a possibility, they haven't seen r-theta yet, as freshman in one of your classes. But if they had college calculus, they should have seen it. All in all, it's still not as central a concept as x-y equations and cartesian coordinates. (Still, important, shouldn't be excised, but not AS important.)
P.s. You know you could just look at some typical textbooks or public high school curricula, to research questions like this. Not a snipe, or to stop from asking the forum also. But...multiple methods...
Neither topic is covered as a Common Core standard as such.
Somewhere in middle school, students learn that a circle cannot be a graph of a function because it fails what we call the Vertical Line Test. (I could look up the exact standard and grade level if you're curious about it.) In Geometry (tenth or eleventh grade typically, but it can vary by district), students learn to calculate the standard and general equations for a circle based on its radius and the coordinate of its center and working backwards from there, but we don't describe it as an implicit function. There are no common standards for pre-calculus at the moment, but people generally cover conics to some level of detail. Implicit differentiation is covered in the AP Calculus standards (both AB and BC), but IME we don't miss the lack of prior knowledge.
When I was in high school 35 years ago, we did cover parametric equations in what was essentially pre-calculus, and I believe differentiating parametric equations came along with studying the Chain Rule in Calc AB. I can easily imagine that some college courses would still cover it, but it is no longer in the AP Calculus standards.
