CLEP College Algebra

additive inverse to a

-a

multiplicative inverse to a (when a does not =0)

1/a

raising expression to a negative exponent

1/b^n

multiplying expressions with powers

x^n + x^m = X^m+n

raising expressions to powers

(x^m)^n = x^mn

diving expressions with powers

x^m/ x^n = x^m-n

fractional exponents

x^1/2 = sq rt. X

x^m/n = n rt. of x^m

multiplying radicals

(sq a)(sq b) = sq ab

factoring diff. of two squares

x^2-y^2 = (x-y)(x+y)

factoring perfect squares

x^2+2xy+y^2=(x+y)^2

x^2-2xy+y^2=(x-y)^2

factoring: sum and difference of 2 cubes

x^3+y^3=(x+y)(x^2-xy+y^2)

x^3-y^3=(x-y)(x^2+xy+y^2)

slope of line (given two points)

m = rise over run y2-y1/x2-x1

equation of line slope-intercept form (given m and y-intercept)

y=mx+b m is slope and b is y intercept

point slope form (given two points)

(y-y1)=m(x-x1)

intercept form (given x and y intercept)

x/a + y/b =1

slope of perpendicular lines

inverses on each other ex. m=b inverse slope is 1/2

discriminant used to determine the number of real solutions

b^2-4ac = > 0 is 2 real solutions

=0 is 1 real solution

< 0 is 0 real or 2 imaginary solutions

logarithm addition

log a + log b = log ab

logarithm subtraction

log a - log b = log a/b

logarithm power rule

log a^b = b log a

arithmetic sequence

an= a1 + (n-1)d

sum of arithmetic sequences

Sn= n/2[2a1+(n-1)d] or

Sn= n/2 (a1+a2)

geometric sequence

an=a1r^n-1 where a1 is the first term, r is the common ratio and n is number of terms

sum of geometric sequences

Sn= [a1(1-r^n)]/ 1-r

sum of infinite geometric sequences

Sn= a1/1-r

a1 is the first term, r is the common ratio, and n is the number of terms

factorial

n! = n(n-1)(n-2)... 3(2)(1)

combination of n choose r

nCr= n!/r!(n-r)!

determinant of 2x2 matrix

a b

c d = ad-bc

find the additive inverse of (expression)

solve the equation for y: y + (expression) = 0

it is the negative of the expression

raise expression to a negative number

1/ (expression)^n

raise to a fractional power

expression^1/n = n rt (expression)

to determine whether a graph is a function

vertical line test

determine whether a graph matches that of an algebraic function

plug x-values into the function to find the value of y, and determine if those points are on the graph

given y=f(x) find the domain

domain will be infinity, infinity with 3 exceptions

1. any value of x that makes the denominator 0

2. any value of x that makes an even root of a negative number

3. any value of x that makes a term into the log of zero or the log of a negative number

graph a line y=mx+b

chose two values for x and find the values of y draw a line between both points

determine whether a graph is odd, even, or neither

even- if the graph is symmetrical with respect to y-axis

odd- if the graph is symmetrical with respect to the origin (I and III and II and IV are mirror images)

given f(x) describe transformations

f(x) + a is a units up

f(x) - a is a units down

f(x-a) is a units to the right

f(x+a) is a units to the left

-f(x) reflects across x-axis

given a graph, determine how many roots

how many times the graph touches/crosses the x-axis

given a polynomial of degree n find the max and min # of roots

max number of real roots is n

if n is even, the min is 0

if n is odd the min is 1

given f(x) determine the roots

set f(x)=0 and factor f(x). set each factor to zero and solve.

find the inverse f^-1 of y = f(x)

interchange x and y and solve for y

given the graph of f(x), determine if the inverse is a function

apply the horizontal line test on f(x)

solve an inequality in the form a< f(x)

a < f(x) and f(x)< b

solve an absolute value equation in the form of |f(x)| = a

f(x)=a and -f(x)=a

solve a system ex.

ax+by=c

dx+ey=f

1. substitution- solve for one variable and plug into the other equation

2. elimination- multiply both equations by numbers that allow you to add the numbers and one variable cancels

solve a system of inequalities graphically

graph each inequality shading the solution regions, the common shaded area is the solution

determine whether a function is a growth or decay

when the equation is in the form of y=a^x the graph is growth if a>1 and decay if 0>a>1

find ln=y

y =e^x, most answers are left in terms of e

LOG b f(x) = n

b^n = f(x)

rational numbers

integers, fractions, terminating decimals, repeating decimals

(a+bi)(a-bi)

i^2 =1

a+bi/ci

multiply by i/i and simplify

given An find the nth term

plug n into the formula for An

given a formula for An, find the partial sum of Sn

if n is small, generate n terms and add them or use the formula for Sn

determine whether a sequence is arithmetic, geometric, or neither

if there is a common difference, it is arithmetic. if there is a common ratio, it is geometric

given a1, d, n, find the nth term

An = a1 + (n-1)d

find the sum of the first n terms of an arithmetic series

Sn= n/2 [2a1+(n-1)d]

or Sn= n/2 (a1+an)

given a1, r and a value for n, find the nth term in a geometric series

an= a1r^(n-1)

find the sum of the first n terms of a geometric series

Sn=[a1(1-r^n)]/1-r

find the sum of an infinite geometric series

if |r|

n!

n!= n(n-1)(n-2)....(3)(2)(1)

nCr

=n!/ r!(n-r)!