Algebra refers to the branch of mathematics where-in we use alphabets and numbers in formulas and equations to represent quantities that are otherwise unknown. For example, if one pen costs 2 \- then using algebra we can say that x pens would cost  2x. Here x would represent the variable that represents the number of pens. So if x = 10 then the cost would be 2 $\times$ x = 2 $\times$ 10 = $20/-.

Algebra in high school or college is usually split into 3 parts. They are: pre-algebra, algebra 1 and algebra 2. Thus, algebra 2 is the third algebra course in high school.

The topics usually covered under algebra 2 are listed below:

1. Equations and inequalities
2. Graphing functions and linear equations
3. System of linear equations
4. Matrices
5. Polynomial and radical expressions
6. Quadratic functions and inequalities
7. Conic sections
8. Polynomial functions
9. Rational expressions
10. Exponential and logarithmic functions
11. Sequences and series
12. Discrete mathematics and probability
13. Trigonometry

Sometimes the topics of conic sections, discrete mathematics, probability and trigonometry are moved out of algebra 2 and included in pre-calculus as well, since they form a precursor to calculus. Each of these topics can be further sub-divided into various sub-topics.




Common core are standards of high quality in academics for math and English. These standards define an outline of what a student should know at the time when he or she completes a specific course or a specific grade. There are common core standards for algebra 2 as well. The detailed concepts covered in algebra 2 under common core standards are as listed below:

1.  Functions: 
     a. Various operations on functions such as addition, subtraction, multiplication and composition. 
     b. Finding of inverse of functions both graphically as well as algebraically.
     c. Special functions such as piecewise, step, absolute value etc and their graphs and transformations along with transformations of standard functions like linear, quadratic, reciprocal etc.
     d. Average rate of change using tables, equations and graphs

2. Systems of inequalities
      a. Solution to inequalities in one variable.
      b. Graphing linear and non-linear inequalities and identifying their solution regions.
      c. Linear system of inequalities, their solutions and applications
      d. Linear programming problems.

3. Quadratics
      a. Assume that all concepts of algebra 1 related to quadratics are thoroughly known and ingressed.
      b. Introduction to imaginary numbers and arithmetic operations on them.
      c. Quadratic equations with complex numbers as solutions.
      d. Graphing of quadratic functions.
      e. Application of quadratic functions.

4. Exponential and logarithmic functions
      a. Exponential models for growth and decay of quantities.
      b. Graphing exponential and logarithmic functions.
      c. Solving exponential and logarithmic equations.
      d. Laws of exponents and laws of logarithms.
      e. Real world applications of exponential and logarithmic functions.

5. Polynomials
      a. Operations on polynomials like multiplication, division, addition and subtraction.
      b. Remainder and factor theorem, identifying zeros (or roots), solving for zeros and graphing polynomial functions.
      c. Methods of division: Synthetic division and long division.

6. Rational and radical functions
      a. Writing, rewriting, and solving simple radical equations with real or imaginary solutions.
      b. Arithmetic operations on radical expressions.
      c. Solving rational equations.
      d. Simplifying radical expressions.

7. Sequences and series
      a. Arithmetic sequence and series.
      b. Geometric sequence and series.
      c. Sum of finite terms of an arithmetic series using formula.
      d. Sum of finite terms of a geometric series using formula.

8. Statistics
      a. Mean, median, mode, range and standard deviation.
      b. Normal distribution, normal distribution curve and area under it.
      c. Simple random sampling.
      d. Process of data generation – step by step.
      e. Margin of error – definition and calculations.
      f. Understanding the meaning of level of significance.

9. Trigonometry
      a. Trigonometry with reference to a right triangle.
      b. Six trigonometric functions of any angle.
      c. Angle measure in radians and length of arc of circle.
      d. Unit circle and its relation to trigonometric ratios.
      e. Fundamental trigonometric identities.
      f. Graphing of trigonometric functions and its application with reference to amplitude, period and median line.


Given below are a few examples of problems that one may encounter when learning algebra 2.

Example 1: Find the inverse of the function given below:

f(x) = $\frac{(x+5)}{(x-2)}$

Solution:

Step 1: Replace the f(x) with y.

y = $\frac{(x+5)}{(x-2)}$

Step 2: Switch the positions of x and y with each other so that all x's are changed to y and the y on the left side becomes x.

x = $\frac{(y+5)}{(y-2)}$

Step 3: Solve this equation for y.

First we cross multiply:

$\frac {x}{1}$ = $\frac{(y+5)}{(y-2)}$

x(y-2) = 1(y+5)

Next we distribute:

xy - 2x = y+5

Collect all y terms on the left side and all other terms on the right side.

xy - y = 2x + 5

Factor out the y from the left side:

y(x - 1) = 2x + 5

Divide both the sides by (x-1)

y = $\frac{(2x+5)}{(x-1)}$

Step 4: Replace the y with f^(-1) (x)

$f^(-1) (x)$ = $\frac{(2x+5)}{(x-1)}$  ←Answer!


Example 2: Solve the following exponential equation:

$3(5^0.2x)$ = 720

Solution:

Step 1: Divide both the sides by 3.

$5^0.2x$ = $\frac{720}{3}$

$5^0.2x$ = 240

Step 2: Take common logarithm on both the sides.

$log⁡(5^0.2x)$ = log⁡240

Step 3: Apply laws of logarithms to get rid of the exponent.

$0.2x(log5)$ = log⁡240

Step 4: Find and substitute the values of log5 and log240.

0.2x(0.6990) = 2.3802

Step 5: Simplify the left side.

0.1398x = 2.3802

Step 6: Divide both the sides by 0.1398 to solve for x.

x = $\frac{2.3802}{0.1398}$

x = 17.03 ←Answer rounded to two decimal places.

Example 3: Find the value of sin and cos of the angle $\frac{17\pi}{6}$.

Solution:

The angle $\frac{17\pi}{6}$ can be written as $\frac{(18\pi-\pi)}{6}$.

$\frac{17\pi}{6}$ 

$\frac{(18\pi - \pi)}{6}$.

$\frac{18\pi}{6}$ - $\frac{\pi}{6}$

=$\frac{3\pi-\pi}{6}$

We know that 3$\pi$ would be the point (-1,0)on the unit circle. Subtracting $\pi$/6 from it would mean that our angle lies in the second quadrant. Thus, our reference angle is $\frac{\pi}{6}$ in Q2.

From our special angles trigonometric values we know that:

sin⁡($\frac{\pi}{6}) $ =   $\frac{1}{2}$

And 

cos⁡($\frac {\pi}{6})$ = $ \frac{\sqrt 3}{2}$

Now, since the angle is in second quadrant, its sine would be positive and cosine would be negative. Thus,

sin⁡($\frac{17\pi}{6}) $ = $\frac{1}{2}$

And

cos⁡($\frac{17\pi}{6})$ = $ \frac{-\sqrt 3}{2}$