Algebra

• Functions
  • Square Root
• Completing squares
• Logarithm
• Complex

quadratic means a variable multiplied by itself “involving the second and no higher power of an unknown quantity or variable.”

Transcendental numbers

\(0^0\) = it is either 1 or undefined, the context depends

Algebraic fractions, \(\frac{50x - 77}{5000}\)

Proportionality, 4y = 8x

Rates of change, \(\frac{f(b) - f(a)}{b - a}\)

long division

• Expression is 25x + 5
• Equation is 25x + 5 = 1000

\(1^0 = 1, 2^0 = 1, N^0 = 1\) (except for 0)

Exponents \(a^b\), a is base, b is exponent or power (multiplication, factoring) algebraic expressions

Quadratic formula If \(ax^2 + bx + c = 0\), then \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Binomial expansions \((x+a)^2 = x^2 + 2ax + a^2\) \((x+a)^3 = x^3 + 3ax^2 + 3a^2x + a^3\)

Absolute value equivalences If \(\lvert ax + b \rvert < c\), then -c < ax + b < c. If \(\lvert ax + b \rvert > c\), then ax + b > c or ax + b < -c.

Distance formulas d = rt (where d=distance, r=rate, and t=time) d = \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of points

Equations of graphs Line: y = mx + b (where m = slope and b = y-intercept) or Ax + By = C Parabola: \(y=ax^2\) + bx + c or y - k = \(a(x - h)^2\) (where (h, k) = the vertex)

Functions

• Domain
• Range

x is called the independent variable and y is called the dependent variable.

Basic form is simply f(x) = y, which is read as “y in a function of x.”

even vs odd function

Exponential and logarithmic functions

Transformation of functions

Compositions

Square Root

def my_sqrt(number):
    """ Custom square root function using the Newton-Raphson method/Heron's method """

    # Condition to exit if the function receives a number less than 0
    if number < 0:
        return "Cannot compute square root of a number less than 0"

    # Acceptable condition, the tolerance can be adjusted
    tolerance = 0.00001
    
    # First guess
    guess = number / 2.0
    
    while abs(guess * guess - number) > tolerance:
        # The formula
        guess = (guess + number / guess) / 2.0
    
    return guess

print(my_sqrt(-1))
print(my_sqrt(0))
print(my_sqrt(2))
print(my_sqrt(4))
print(my_sqrt(16))

Completing squares

Logarithm

To solve for a in a formula: \(log_a (b) = \frac{log_x (b)}{log_x (a)}\)

The value of \(log_2 3\) (w/o a calculator)

Start with upper bound:

1. x = \(log_2 3\)
2. This means: \(2^x = 3\)
3. \[(2^x)^3 = 3^3\]
4. 27 is less than 32 which is \(2^5\).
5. This means 3x is less than 5.
6. x < \(\frac{5}{3}\) then x < 1 \(\frac{2}{3}\)

Now do lower bound:

1. \[2^x = 3\]
2. \[(2^x)^2 = 3^2\]
3. 9 is greater than 8 which is \(2^3\).
4. This means 2x is greater than 3.
5. x > 3/2 then x > 1 \(\frac{1}{2}\)

We get \(1 \frac{1}{2} < x < 1 \frac{2}{3}\).

Complex

Imaginary numbers = i = \(\sqrt{-1}\)

Has the form: a+bi