csc x = 1/sin x
sec x = 1/cos x
cot x = 1/tan x
sin (x/2) = radical((1 - cos x) /2)
cos (x/2) = radical((1 + cos x) / 2)
tan (x/2) = (sin x / 1 + cos x)
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.
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.
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.
tan x = sinx/cosx
cot x = cosx/sinx
cos2x = cos2x - sin2x
sin2x = (2sinx)(cosx)
tan2x = (2tanx / 1 - tan2x)
f(x) = a graph is a horizontal line
f(x) = x points on graph have coordinates (a, a)
f(x) = x2 graph is U-shaped
f(x) = x3 graph is symmetric about the origin
square root function
f(x) = sqrt(x) graph is in first quadrant
f(x) = 1/x graph has two branches
absolute value function
f(x) = │x│ graph is V-shaped
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
y = ln x
y = bx