Understanding Algebra 1 and Algebra 2

Understanding Algebra 1 and Algebra 2

What is Algebra?

Algebra is a branch of math where we use letters like x and y to represent unknown numbers. Instead of just working with numbers, we write and solve equations like:

x + 3 = 7

It's like solving a math puzzle to find out what x is!

History of Algebra + A Fun Fact

Algebra started around the 800s AD by a Persian mathematician named Al-Khwarizmi. His famous book was titled Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala. The word Al-Jabr turned into Algebra in English! His name also gave birth to the word algorithm, which we now use in computers.

Fun Fact:

Even in ancient Egypt, people used algebra-like thinking to split things evenly or solve riddles like:

One person has 2 more apples than the other, and together they have 10. How many apples does each have?

Algebra 1 (Typically 7th–9th grade)

Variables and Expressions

Example Problems:

- 3x + 5

- x - 7

- 2y * 4

- (x + 2)(x - 3)

- 5(x + 1)

Real-Life Applications:

Buying x candies for 3 dollars each → Total cost = 3x

Earning 10 dollars per hour for x hours → Pay = 10x

Equations and Inequalities

Example Problems:

- x + 3 = 7

- 2x - 4 = 10

- x > 5

- 3x + 2 <= 11

- 5x - 1 >= 14

Real-Life Applications:

You're 3 years younger than someone who is 15 → x + 3 = 15

You need 70 or more points to pass a test → x >= 70

Functions

Example Problems:

- f(x) = x + 2

- f(x) = 3x

- f(x) = x^2

- f(x) = x - 4

- f(x) = 2x + 5

Real-Life Applications:

Vending machine: You put in x dollars, and get a drink → Output depends on x

Bus fare: 1.20 base + 0.10 per km → Cost function

Linear Functions and Graphs

Example Problems:

- y = 2x + 3

- y = -x + 1

- y = 0.5x - 4

- y = 3x

- y = x - 2

Real-Life Applications:

Taxi fares (base price + per mile)

Hourly wages → Graph shows time vs money earned

Systems of Equations

Example Problems:

- x + y = 10, x - y = 2

- 2x + y = 8, x - y = 3

- x + 2y = 7, 3x - y = 5

- x = y + 4, 2x - y = 9

- x - 3y = 1, 4x + y = 11

Real-Life Applications:

Figuring out prices of pizza and soda given two different combos

When two people walking from different places meet

Algebra 2 (Typically 9th–11th grade)

Polynomials and Factoring

Example Problems:

- x^2 + 5x + 6

- 4x^2 - 9 -> (2x + 3)(2x - 3)

- x^3 + 2x^2 - x - 2

- x^2 - 4x + 4

- (x + 1)(x - 2)(x + 3)

Real-Life Applications:

Calculating area of garden plots

Designing curved ramps in construction

Imaginary Numbers

Example Problems:

- i = sqrt(-1)

- i^2 = -1

- (2 + 3i) + (1 - i)

- (4i)(2 - i)

- |3 + 4i| = sqrt(3^2 + 4^2) = 5

Real-Life Applications:

Electric circuits use imaginary numbers for phase shifts

Computer graphics and quantum physics

Quadratic Equations and Parabolas

Example Problems:

- y = x^2 + 2x + 1

- y = -x^2 + 4x

- x^2 = 9

- x^2 - 5x + 6 = 0

- x^2 + x - 12 = 0

Real-Life Applications:

A basketball’s arc when shot

Firework trajectories

Bridge arch shapes

Exponential and Logarithmic Functions

Example Problems:

- y = 2^x

- y = 3^x + 1

- log_2(8) = 3

- log_10(100) = 2

- y = log_5(x - 1)

Real-Life Applications:

Virus spread patterns (COVID)

Bank interest growth

Password encryption and decryption

Sequences and Series

Example Problems:

- 2, 4, 6, 8, ...

- 1, 2, 4, 8, 16, ...

- a_n = 2n

- sum(n=1 to 4) = 1 + 2 + 3 + 4 = 10

- a_n = 3 + (n-1) * 5

Real-Life Applications:

Saving money over months

College fund planning

Repeating work patterns or shifts