Algebra 2?

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Kets
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Algebra 2?

Postby Kets » Mon Aug 10, 2009 7:04 pm UTC

I was wondering what I would have to learn in Algebra 2, and would like to try to memorise some of it before the school year starts.

Tac-Tics
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Re: Algebra 2?

Postby Tac-Tics » Mon Aug 10, 2009 7:17 pm UTC

Algebra 2 doesn't have a universally recognized course guide. It depends on your school.

Generally speaking, though, it is a natural continuity after algebra 1. In my second year algebra course, the only thing I remember learning was the complex number system and polar coordinates. Expect lots of polynomial functions. Recognizing particular graph behaviors (number of zeros, number of critical points, etc) match up with the degrees of a polynomial.

It might also blend together with notions in precalculus. Developing a graphical intuition over the notions of an asymptote, the notion of a limit of a graph at points where it is either undefined or at points infinitely far away. There might be some logarithms and exponential functions in there too.

Kets
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Re: Algebra 2?

Postby Kets » Mon Aug 10, 2009 7:42 pm UTC

What is a zero / a critial point?

achan1058
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Re: Algebra 2?

Postby achan1058 » Mon Aug 10, 2009 9:13 pm UTC

A zero of a function is the points where the function is 0. ex. If [imath]f(x)=x^2-1[/imath], then the zeroes will be [imath]x=1,-1[/imath]. A critical point is where the function reaches a maximum/mimimum. ex. x=0.

Anyways, a note of mathematics in general. It is not about memorization. The way you can tell good students from bad students is that good students memorize less. (yet get better grades) You want to understand the material, and know how they connect to each other.

Kets
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Re: Algebra 2?

Postby Kets » Tue Aug 11, 2009 6:08 pm UTC

Thank you for the explanation, and I appreciate the advice.

Can you also explain what asymptotes and logarithms are?

achan1058
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Re: Algebra 2?

Postby achan1058 » Tue Aug 11, 2009 8:33 pm UTC

If you draw a function on a piece of graph paper, an asymptote is some kind of line that a function gets closer and closer to, but never reaches. Logarithm is the inverse operation of exponentiation.

Anyways, if you want more information, wikipedia is always a good start. They are very accurate when it comes down to mathematics and science.

pingu
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Re: Algebra 2?

Postby pingu » Tue Aug 11, 2009 11:51 pm UTC

Just for clarification, it should be known that asymptotes can be intersected, even infinitely many times.

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lu6cifer
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Re: Algebra 2?

Postby lu6cifer » Wed Aug 12, 2009 10:25 am UTC

From what I remember, here's a rough outline of my Alg2 course:

Learning about different properties and field axioms (distributive property, associative, etc...)
Solving linear equations(via substitution, linear addition, matrices), graphing linear equations, reading graphs
Inequalities, graphing inequalities
Quadratic equations and graphing them; asymptotes and removable discontinuities
Polynomial stuff..aka polynomial division/factoring review/other
Exponents review, then logarithms--as stated before, a logarithm is the inverse of an exponent

Example: 2^5 = 32: Base 2 to the power of 5 is 32. So, log(32), base 2, equals 5.

Then conic sections, I believe (parabolas, hyperbolas, ellipses, circles....honestly, the most annoying part of algebra 2)
Then series and sequences, which was really just a segue into beginning sigma notation

I probably left some stuff out, but most of it's there.
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Alpha Omicron
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Re: Algebra 2?

Postby Alpha Omicron » Wed Aug 12, 2009 6:45 pm UTC

I'd like to strongly second achan1058's comment about memorisation. If you have to memorise something, you don't understand why it is that way.
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Tac-Tics
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Re: Algebra 2?

Postby Tac-Tics » Wed Aug 12, 2009 8:09 pm UTC

Kets wrote:Can you also explain what [...] logarithms are?


The logarithm takes a number and maps it to its order of magnitude. It's usually denoted "log".

Each log has a "base". The most common bases are 2, e, and 10. When the base is e, we call it the "natural" log (and we denote it "ln" instead of "log").

One plus the integer part of the log (base 10) tells you the number of digits in the number.

One plus the integer part of the log base 2 tells you the number of bits required to encode the number.

The log of a number in scientific notation is approximately equal to the exponential part. So log(5.2 * 10^24) ~ 24.

Logarithms are the basis for the slide rule, a mechanical calculator that was used extensively up until the digital calculator became available.

Log(x) is undefined for x <= 0. The logarithm can be extended to work on negative numbers if we allow for complex numbers.

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majikthise
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Re: Algebra 2?

Postby majikthise » Wed Aug 12, 2009 8:56 pm UTC

Tac-Tics wrote:The logarithm takes a number and maps it to its order of magnitude. It's usually denoted "log".

Why isn't it always introduced to school kids in this way? Seems much easier to grasp at first than the usual definition.
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Kow
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Re: Algebra 2?

Postby Kow » Thu Aug 13, 2009 1:28 am UTC

Alpha Omicron wrote:I'd like to strongly second achan1058's comment about memorisation. If you have to memorise something, you don't understand why it is that way.

That's one thing I always hated about busywork homework. If I understand the concept, I don't need to practice it! I've done poorly in many a math class for that reason, despite knowing the material possibly the best in the class.
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Mindor
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Re: Algebra 2?

Postby Mindor » Sat Aug 15, 2009 8:21 pm UTC

Kow wrote:
Alpha Omicron wrote:I'd like to strongly second achan1058's comment about memorisation. If you have to memorise something, you don't understand why it is that way.

That's one thing I always hated about busywork homework. If I understand the concept, I don't need to practice it! I've done poorly in many a math class for that reason, despite knowing the material possibly the best in the class.


If you mean you were graded poorly because your grade depended on completion of homework then I've faced this too.
On the other hand, understanding and knowing material are different.
Understanding stays with you much longer, and allows you to apply the subject more broadly.
I couldn't tell you how many times I've sat for a test and looked at a problem and wasn't sure which of many equations applied to it, but was able to quickly re-derive what was needed because I really understood the subject. If I knew the material I wouldn't need to figure it out during the test. When I've had the time to actually do the homework, I've never faced this problem. So even though it feels like busy work, it does help ingrain the specifics for ease of use later.
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porcupine
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Re: Algebra 2?

Postby porcupine » Sun Aug 16, 2009 5:45 am UTC

I agree with those who say it's about understanding and not memorization. My trig teacher completely ignored the book for most of the year, instead showing and explaining to us how each formula is derived. It was very helpful to me, at one point I showed a proof for using one long equation to find the length of a side of a triangle that included the the low of sines and cosines, and got five extra points for it. :) It was annoying to type in an equation three lines long, but it was quicker than using one formula, then saving the result to a variable, then using another one. I my not have been the fastest at math in my class, but at parent/teacher conferences my teacher told me that I was one of the few people that really understood what was going on.

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Qaanol
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Re: Algebra 2?

Postby Qaanol » Sun Aug 16, 2009 5:54 pm UTC

The logarithm is the exponent.
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Tac-Tics
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Re: Algebra 2?

Postby Tac-Tics » Mon Aug 17, 2009 2:23 pm UTC

Qaanol wrote:The logarithm is the exponent.


The logarithm is the inverse of the exponent.....

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Alpha Omicron
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Re: Algebra 2?

Postby Alpha Omicron » Mon Aug 17, 2009 6:32 pm UTC

Tac-Tics wrote:
Qaanol wrote:The logarithm is the exponent.
The logarithm is the inverse of the exponent.....
The process of taking a logarithm is the inverse of taking an exponent. But the resulting logarithm is an exponent. Think about it.
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Tac-Tics
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Re: Algebra 2?

Postby Tac-Tics » Mon Aug 17, 2009 7:51 pm UTC

Alpha Omicron wrote:
Tac-Tics wrote:
Qaanol wrote:The logarithm is the exponent.
The logarithm is the inverse of the exponent.....
The process of taking a logarithm is the inverse of taking an exponent. But the resulting logarithm is an exponent. Think about it.


That's not the whole story. If you're going to say something like that, word it so your statement is true.

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intimidat0r
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Re: Algebra 2?

Postby intimidat0r » Tue Aug 18, 2009 5:12 am UTC

The natural logarithm function, ln(x), is the inverse of the exponential function, exp(x) = e^x.

Similarly, the function log(x) is the inverse of the function 10^x.

For example, if you have 10^x = 1000, then x = log(1000) = 3.
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Kalathalan
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Re: Algebra 2?

Postby Kalathalan » Thu Aug 20, 2009 6:47 am UTC

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Last edited by Kalathalan on Mon Jun 06, 2016 8:07 am UTC, edited 1 time in total.

dean.menezes
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Re: Algebra 2?

Postby dean.menezes » Thu Aug 20, 2009 1:55 pm UTC

achan1058 wrote:A zero of a function is the points where the function is 0. ex. If [imath]f(x)=x^2-1[/imath], then the zeroes will be [imath]x=1,-1[/imath]. A critical point is where the function reaches a maximum/mimimum. ex. x=0.

Anyways, a note of mathematics in general. It is not about memorization. The way you can tell good students from bad students is that good students memorize less. (yet get better grades) You want to understand the material, and know how they connect to each other.

Technically, a critical point is any point where the derivative is zero or where the derivative does not exist.
Also, just because the derivative is zero does not mean it is necessarily a minimum or a maximum -- it could be a point of inflection

Token
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Re: Algebra 2?

Postby Token » Thu Aug 20, 2009 5:53 pm UTC

dean.menezes wrote:Also, just because the derivative is zero does not mean it is necessarily a minimum or a maximum -- it could be a point of inflection

Or merely a local minimum or maximum. [/pedant]
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