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I'm about to start teaching a calculus I class next week and I was wondering what I can expect from my students. I'm a Brit teaching in the US so I am unfamiliar with the system.

I am hoping that they can:

1) Factor simple quadratic (cubic?) polynomials, complete the square, use the quadratic formula and know the difference between squares factorization.

2) Know the basic trig functions and their elementary definition (e.g. sin = opposite/hypotenuse)

3) Have some grasp of the notion of a function.

Is this a reasonable collection of expectations?

Oh and I plan to use this to construct a self-diagnostic review quiz to give the kids.

Later additions:

4) Laws of logarithms and of exponentials.

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    $\begingroup$ You can always click on the "algebra-precalculus" tag and see what roams around in MSE. $\endgroup$ – Pedro Tamaroff Aug 22 '13 at 21:22
  • $\begingroup$ Going off of Peter's idea, the tag-wiki for [algebra-precalculus] describes this body of math(s) as follows: "linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics." It seems that you're on track, and there are a few more suggestions for topics to boot. $\endgroup$ – Omnomnomnom Aug 22 '13 at 21:31
  • $\begingroup$ Maybe combinatorics too? $\endgroup$ – Hawk Aug 22 '13 at 21:33
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I would recommend looking at various syllabi from different schools to see the scope of what they are doing.

For example:

If you summarize them, you would have something that looks like the list you wrote, or something like:

  • The analytical geometry of the Cartesian plane (distance, slope, lines, circles, parabolas, ellipses, hyperbolas, their intercepts, vertices, centers, radii, major and minor axes, foci, and tangents),
  • The algebra of polynomials and rational functions (zeros, factoring, remainder theorem, asymptotes, intercepts, sketching, multiplicity, complex roots, algebra of the complex numbers, inverses of polynomial and rational functions; one-to-one, onto functions; composition)
  • Exponential functions and logarithms (the laws of exponents and logarithms, solving exponential and logarithmic equations, applications to population, radioactive decay, Newton's law of heat transfer, etc)
  • Circular functions and their inverses (definitions of the six circular functions, special values, modeling with the sine and cosine, right triangle trigonometry, trigonometric identities, solving trigonometric equations, the inverse trigonometric functions)
  • Systems of 2 non-linear equations in two unknowns, especially pairs of linked quadratic equations. (This is a skill needed in calculus 3.)

I would also recommend looking at various books used by schools and considering a Computer Algebra System as part of the curriculum. It is possible to find additional materials as part of the International Opencourseware project, like MIT.

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Yes, those all sound like reasonable expectations for a typical American precalculus course. Students should also have basic familiarity with exponential functions and logarithms, as well as being able to solve some (easy) higher-degree polynomials using tools like the rational roots test.

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  • $\begingroup$ ...all except factorising "simple" cubics. It depends how one defines "simple". If the cubics have small integer roots then trial-and-error, the remainder theorem and some polynomial division should allow them to find the factorisation. In general, it would be quite hopeless. $\endgroup$ – Fly by Night Aug 22 '13 at 21:55

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