Precal Formulas, Rules, Etc.

Front 1

properties of absolute values

Back 1

(i) |a|=|-a|
(ii) |a|=0 if and only if a=0
(iii) |ab|=|a||b|
(iv) |a/b|=|a|/|b|, b≠0
(v) |a+b|≤|a|+|b| (triangle inequality)
(vi) |x|<a if and only if -a<x<a
(vii) |x|>a if and only if x>a or x<-a

Front 2

distance between two numbers

Back 2

d(a,b)=|b-a|

Front 3

midpoint

Back 3

m=(a+b)/2

Front 4

factorizations worth knowing

Back 4

difference of two squares:
a²-b²=(a-b)(a+b)
difference of two cubes:
a³-b³=(a-b)(a²+ab+b²)
sum of two cubes:
a³+b³=(a+b)(a²-ab+b²)

Front 5

binomial expansions worth knowing

Back 5

n=2: (a+b)²=a²+2ab+b²
n=3: (a+b)³=a³+3a²b+3ab²+b³

Front 6

vertical and horizontal shifts

Back 6

suppose y=f(x) is a function and c is a positive constant.
(i) y=f(x)+c shifts vertically up c units;
(ii) y=f(x)-c shifts vertically down c units;
(iii) y=f(x+c) shifts horizontally to the left c units;
(iv) y=f(x-c) shifts horizontally to the right c units.

Front 7

reflections

Back 7

(i) y = -f(x) is the graph of f reflected in the x-axis;
(ii) y =f(-x) is the graph reflected in the y-axis.

Front 8

vertical stretches and compressions

Back 8

suppose y = f(x) is a function and c is a positive constant. Then the graph of y = cf(x) is the graph of f
(i) vertically stretched by a factor of c units if c>1;
(ii) vertically compressed by a factor of c units if 0<c<1.