# Extension activities in Algebra II

I'm in Algebra II this year, and I have to admit, it's kind of boring. The only new thing we've touched on so far this year is how to graph piecewise functions, and those are really easy to graph.

As it is, since the teacher puts the problems for the week on the backboard, I work ahead based on the homework for the next couple of days (there were a couple of days where I was the full week ahead, so I just read in the back of the class).

While I'm investigating options to generally accelerate the class (maybe try to finish it in a semester instead of two?) I know the likelihood of those options working out is very, very low, so I'm asking this question.

What sorts of extension or substitute activities could be done based on the Algebra II curriculum in the United States? For whatever it's worth, I've done a decent amount of math on my own - some linear algebra and calculus, a general awareness of some of the topics in abstract algebra, and a heavy interest in physics.

Edit [12.16.17]: A write-up concerning a few of the problems below was accepted for publication in NCTM's grades 8-14 journal, The Mathematics Teacher. It is slated for publication in Fall 2018, but a pre-pre-print can be found here (RG link to PDF, 8pp).

I taught two sections of Algebra 2 (ninth and tenth grade, US) last year. The materials we use are internally created, and while there is not (yet?) a public repository that makes all of them freely available, I paste below a problem set that I personally wrote for students to complete. I'm not sure if this is the sort of thing that you are looking for, but the problems are (i) related to Algebra 2; (ii) original hence cannot be found elsewhere; and (iii) nonstandard and, I think, reasonably challenging.

1) As before, let us write $i$ to denote $\sqrt{-1}$.

a) Is $\sqrt{5+i}$ a real number or an imaginary number? Explain how you know.

Hint: Find a polynomial $f$ for which $\sqrt{5+i}$ is a root, i.e., for which $f(\sqrt{5 + i}) = 0$. What does the graph of $f$ indicate about all of the roots of $f$?

b) Is $\sqrt{i}$ a real number or an imaginary number? Explain how you know, using a method similar to your approach in 1(a).

c) Let $\alpha$ be the number $\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$. Carefully calculate $\alpha^2$. Does this support or contradict your answer in part (b)? Explain!

2) Suppose $f$ is a polynomial for which the sum of the coefficients is $0$.

a) Give an example of a cubic polynomial $f$ that satisfies the above criterion, and find all of its roots.

b) Explain why any polynomial whose coefficients sum to $0$ has, in its graph, at least one $x$-intercept.

c) Give an example of a polynomial $f$ with degree $5$, whose coefficients sum to $0$, which has only one $x$-intercept. Explain clearly how you know that your example polynomial has only one $x$-intercept.

3) Suppose $f$ is the following piecewise defined function:

$f(x) =$

$$2x \text{ when } x \leq -3;$$

$$(x-h)^2 - 6 \text{ when } -3 < x \leq 0; \text{ and}$$

$$4x-h \text{ when } x > 0$$

a) For what value of $h$ is the piecewise defined function above continuous (i.e., the graph has no "jumps")?

b) Using the value of $h$ that you found in 3(a), sketch a graph of $f$.

c) Does $f$ have an inverse function? If not, explain clearly why not; if so, explain why (but you do not need to find its inverse).

4) Let $f$ be the polynomial defined by $f(x) = -x^5 + Ax^4 + Bx^3 + Cx^2 + Dx + 2$, where each of $A, B, C,$ and $D$ is an integer. Suppose that the graph of $f$ has at least four distinct $x$-intercepts.

Explain clearly why $f$ must have at least two irrational roots.

5) Let $f$ be a quadratic polynomial that can be written in either vertex form or standard form, i.e.,

$$f(x) = a(x-h)^2 + k \text{ or } f(x) = ax^2 + bx + c$$

Suppose further that the domain of $f$ is $[s, \infty)$ for $s \in \mathbb{R}$ minimal such that $f$ is invertible.

a) Find the inverse of $f$ when $f$ is written in vertex form.

b) Find the inverse of $f$ when $f$ is written in standard form.

c) What is $s$, and how do you know?

You might consider look at Exeter's books of math problems or the Art of Problem Solving texts. They are targeted at students around your age but look very different from your standard Algebra II presentation.

My advice is to accelerate. Just think you are better off moving on and continuing your math education since you feel you are ready. Get through the material and start working on trig and pre-calc. If you want to come back and enrich at some later time, so be it. But just think you will be better off moving on.

Personal experience: When I went to school, it was normal for Algebra 2 to be combined with Trig (i.e. 3 semesters in 2) for the kids on the GT track (who had taken algebra 1 in 8th grade.) Also, I did the Alg2/trig in summer before 10th so got more accellerated. (this isn't to say yeah me, but that if you sense you can handle the acceleration, go ahead and power through the material and move on to a real precalc course).

• I would add that you will get plenty of practice at logarithms and exponent rules and fractions of polynomials and the like in analytic geometry (curve tracing) and calculus. And physics and chemistry. So, if you have basic mastery for now (A to A+ level), I would not worry about grinding algebra 2 into the dust for practice. Oct 12 '17 at 16:16

Perhaps you would find this extends Algebra II in interesting and challenging directions:

Kendig, Keith. A guide to plane algebraic curves. Vol. 46. MAA, 2011. (MAA link.)

From the publisher's summary:

"a friendly introduction to plane algebraic curves. It emphasizes geometry and intuition, and the presentation is kept concrete. You'll find an abundance of pictures and examples to help develop your intuition about the subject, which is so basic to understanding and asking fruitful questions."

An 8-petal rose morphing to a Lissajous figure [p.19].