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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.

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  • $\begingroup$ It's not really obvious to me what it would mean to "learn implicit equations." Knowing that $x^2+y^2-r^2=0$ is the equation of a circle is more a matter of understanding the Pythagorean theorem along with understanding the notion of Cartesian geometry. The implicit function theorem is a topic that my community college teaches in first-semester calculus, but I don't think it's necessary to know the implicit function theorem in order to interpret this equation. $\endgroup$ – Ben Crowell Nov 2 at 21:34
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    $\begingroup$ The idea that a curve would naturally be given by $y=f(x)$ is more like something that students would pick up in a first-year college calculus course that included some analytic geometry. For high school students, I don't think that's an idea that has ever been planted in their brains, so they don't need to make any transition. An implicit equation should seem at least as natural. I think issue is more like how much exposure they've had to the Cartesian approach. I think the answer is that they get their first exposure to this in a precalculus course, which might be in 11th or 12th grade. $\endgroup$ – Ben Crowell Nov 2 at 21:54
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    $\begingroup$ Are you asking when they learn or when they are taught? $\endgroup$ – shoover Nov 2 at 21:56
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    $\begingroup$ In my experience: parametric equations are not taught until calculus. $\endgroup$ – Gerald Edgar Nov 3 at 11:39
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    $\begingroup$ @Gerald Edgar: parametric equations are not taught until calculus --- They are covered in some U.S. high school precalculus courses, but this is quite rare. Nearly all U.S. high school students will not see parametric equations until BC calculus, the higher level of AP calculus that only about $3.8\%$ of U.S. high school graduates currently take each year ($140,000$ divided by $3.7$ million). $\endgroup$ – Dave L Renfro Nov 3 at 18:06
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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.

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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.

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  • $\begingroup$ in india students at 11th or 10th grade know it $\endgroup$ – Aditya Dwivedi Nov 17 at 7:10
  • $\begingroup$ @AdityaDwivedi Interesting! $\endgroup$ – Nick C Nov 17 at 14:11
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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.

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  • $\begingroup$ Thanks, Matthew. So much of computer graphics depends on parametric representations, 'tis a shame it is not more central in the curricula. $\endgroup$ – Joseph O'Rourke Nov 5 at 20:10

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