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Algebra Final by

Reciprocal Identities

csc x = 1/sin x
sec x = 1/cos x
cot x = 1/tan x

Half-angle identities

sin (x/2) = radical((1 - cos x) /2)
cos (x/2) = radical((1 + cos x) / 2)
tan (x/2) = (sin x / 1 + cos x)

Transf­orm­ations

Tran­sla­tions

g(x) = f(x) + k is the graph of f(x) translated k units up when k > 0 and k units down when k < 0.

g(x) = f(x − h) is the graph of f(x) translated h units right when h > 0 and h units left when h < 0.


Refl­ect­ions

g(x) = -f(x) is the graph of f(x) reflected in the x-axis.
g(x) = f(-x) is the graph of f(x) reflected in the y-axis.

Dila­tions

g(x) = a · f(x) is the graph of f(x) expanded vertically if a > 1 and compressed vertically if 0 < a < 1.

g(x) = f(ax) is the graph of f(x) compressed horizo­ntally if a > 1 and expanded horizo­ntally if 0 < a < 1.

Domain and range

Doma­in: The domain of a function is the set of all possible input values (often the "­x" variable), which produce a valid output from a particular function. It is the set of all real numbers for which a function is mathem­ati­cally defined.

Range: The range is the set of all possible output values (usually the variable y, or sometimes expressed as f(x)), which result from using a particular function.
 

Cotang­ent­/Ta­ngent

tan x = sinx/cosx
cot x = cosx/sinx

Double­-angle identities

cos2x = cos2x - sin2x
sin2x = (2sinx­)(cosx)
tan2x = (2tanx / 1 - tan2x)

Parent functions

constant function
f(x) = a graph is a horizontal line

identity function
f(x) = x points on graph have coordi­nates (a, a)

quadratic function
f(x) = x2 graph is U-shaped

cubic function
f(x) = x3 graph is symmetric about the origin

square root function
f(x) = sqrt(x) graph is in first quadrant

reciprocal function
f(x) = 1/x graph has two branches

absolute value function
f(x) = │x│ graph is V-shaped
 

Pythag­orean Identities

sin2x + cos2x = 1
tan2x + 1 = sec2x
1 + cot2x = csc2x
You can convert the first identity into the second and third by dividing both sides by cos2x or sin2x.

Expone­ntial and logari­thmic

Logari­thmic
y = ln x

Expone­ntial
y = bx

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