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Advanced Functions Unit 1

Polynomial Functions

Quintic

f(x) = ax^5

Quartic

f(x) = ax^4

Cubic

f(x) = ax^3

Quadratic

f(x) = ax^2

Linear

f(x) = ax

Odd Degree (1, 3, 5, 7, 9...)

*+ 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

Even Degree (2, 4, 6, 8...)

*+ 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

Polynomial Functions in general form:

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

Algebraically determining even and odd functions:

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

Factoring

Sum of Cubes

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

Difference of Cubes

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

Difference of Squares

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

Factoring less simple trinomials

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

Factoring Simple Trinomials

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

Common Factoring

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

Average Rate of Change

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

Instantaneous Rate of Change

- 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

Finite Differences

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)

- 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)

Line/Point Symmetry

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

Subtopic

Line x = a, splits graph into 2 perfect halfs