Advanced Functions Unit 1

Line x = a, splits graph into 2 perfect halfs

Point symmetry about a point (a,b):
- any random point can be rotated 180 degrees about this point of symmetry and match up with another point on the graph

- Formula = nth = a(n!)
- nth: stands for the value of the nth term
- a: stands for the leading coefficient of the function
- n!: is the degree of the function (factorial)

Terms:
- Order: the exponent each factor is raised to
- Order 1 = line through the point
- Order 2 = line bounces back off point
- Order 3 = line "swerves" through the point (looks like x^3 graph)

- Tangent at 1 point on a graph
- Use the "Difference Quotient" formula
- delta y / delta x = (f(a + h) - f(a)) / h
- x = a !!!!!!!!!! ***VERY IMPORTANT***
- plug x in for a when x is given
- h = 0.0001, resembles point very close to x

(f(x2) - f(x1)) / (x2 - x1)
- essentially the slope between 2 given points

Common Factoring

f(x) = ax^2 + bx (factor out commonality in each term)

Factoring Simple Trinomials

f(x) = x^2 + bx + c (where a = 1)

Factoring less simple trinomials

f(x) = ax^2 + bx +c (where a =/ 1) (DON'T FORGET TO GROUP FACTOR)

Difference of Squares

f(x) = a^2 - b^2 = (a+b)(a-b)

Difference of Cubes

f(x) = a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Sum of Cubes

f(x) = a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Odd: f(-x) = -f(x)
* sub (-x) into the function to determine
Even: f(-x) = f(x)
* sub (-x) into the function to determine

f(x) = a(nx)^n + a(n-1)x^(n-1) + a(n-2)x^(n-2) + ... a(3)x^3 + a(2)x^2 + a(1)x + a

Notes: - Arranged in descending order of power
- Exponents must be + whole #'s
- Coefficients must be real #'s
- L.C. = Leading Coefficient
- Degree = highest exponent value

*+ L.C.: As x approaches +/- infinity, f(x) approaches infinity (Q2 -> Q1)
*- L.C.: As x approaches +/- infinity, f(x) approaches negative infinity (Q3 -> Q4)
* have same end behaviors (in the f(x) region)
* Has at most n-1 turning points (n = degree of function)
* Has up to n x intercepts

*+ L.C.: As x approaches infinity, f(x) approaches infinity. As x approaches - infinity, f(x) approaches - infinity (Q3 -> Q1)
*- L.C.: As x approaches infinity, f(x) approaches - infinity. As x approaches - infinity, f(x) approaches infinity (Q2 -> Q4)
* have opposite end behaviors (in the f(x) region)
* Has at most n-1 turning points (n = degree of function)
* Has 1 minimum, up to n x intercepts

Linear

f(x) = ax

Quadratic

f(x) = ax^2

Cubic

f(x) = ax^3

Quartic

f(x) = ax^4

Quintic

f(x) = ax^5