Toolkit of Functions

Toolkit of Functions

Students should know the basic shape of these functions and be able to graph their transformations without the assistance of a calculator.

Constant f(x) = a

Identity

f(x) = x

Cubic f(x) = x3

Square Root f(x) = x

Absolute Value f(x) = | x |

Reciprocal

f(x)

=

1 x

Greatest Integer f(x) = [ x ]

Exponential

f(x) = a x

Quadratic f(x) = x2

Logarithmic

f(x) = ln x

Trig Functions f (x ) sin x

f (x ) cos x

f (x ) tan x

Polynomial Functions:

A function P is called a polynomial if P (x ) anx n an 1x n 1 ... a2x 2 a1x a0 Where n is a nonnegative integer and the numbers a0, a1, a2, ... an are constants.

Even degree Leading coefficient sign

Odd degree Leading coefficient sign

Positive

Negative

Positive Negative

Number of roots equals the degree of the polynomial. Number of x intercepts is less than or equal to the degree. Number of "turns" is less than or equal to (degree ? 1).

Formulas and Identities

Trig Formulas:

Arc Length of a circle:

L = r

Area of a sector of a circle:Area = 1 r 2

2

or L = d 2 r 360

or Area = d r 2 360

Solving parts of a triangle:

Law of Sines:

a

b

c

sin A sin B sin C

Law of Cosines:

a 2 b2 c2 2bc cosA b2 a 2 c2 2ac cos B c2 a 2 b2 2ab cosC

Area of a Triangle: Area = 1 bc sinA 2

or Area = 1 ac sinB or Area = 1 ab sinC

2

2

Heron's formula : Area = s(s a)(s b)(s c) , where s = semi perimeter

Ambiguous Case:

is acute Compute: alt adj sin

opp < alt opp = alt opp>adj

No triangle 1 triangle (right) 1 triangle

is obtuse or right

opp adj No triangle opp > adj 1 triangle

alt ................
................

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