Algebra
Dividing Fractions
Invert (reciprocal) the second fraction and multiply
factors
Numbers that are multiplied together to get a product ie. factors of 120= 60x2= 30x2x2= 15x2x2x2= 5x3x2x2x2 - prime factorization: is the simplest factors of a whole number ie. 28 = 7.2.2
irrational numbers
Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating.
Order of Operations (PEMDAS)
Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction *addition and subtraction: whichever comes first left to right
ratios
a comparison of two quantities using division -can be used to solve lots of word math problems ie 9:2 = F:S therefore when cross multiply 9S = 2F, and can be combined with other ratio to find the variable
Quadratic equations
a function that has the variable raised to the second power, in the form: f(x) = ax² + bx + c = 0, a≠0. *2x^2- 6x+ 4 =0 can be divided by 2 to simplify -> x^2- 3x+ 2=0, (x-1) (x-2) = 0
scientific notation
a method of expressing a quantity as a number multiplied by 10 to the appropriate power 0.000295 = 2.95 x 10^-4 0.00000001 = 1x 10^-8
square root
a number that when multiplied by itself equals a given number √144 = 144^1/2 = +12 or -12 *principle square root: only + square root in the solution = +12 ie. √.0081 = √81x10^-4 = 9x10^-2 or 0.09
cube root
a number that when multiplied three times equals a given number 3√216 = +6 and -6
mixed fraction
a whole number and a fraction. Example 1 1/2 1. add whole numbers first 2. converts to improper fraction *for simple mixed fractions, can just multiply right away ie. 2 1/4 = 4x2+1= 9/4 *or 2 3/4 + 3 1/3 = 5 + 3/4 + 1/3 and find common Dom and solve from there
polynomials
an expression of more than 2 algebraic terms *always try to notice the greatest common factor (GCF) *x will always be more than 1 answer ie. x={0,-3,2} ie. solve for x for 2x^3 +10x^2 +12x = 0 . GCF =2x 2x(x^2+5x+6)= 0 2x (x+2) (x+3)= 0 2x= 0, x+2= 0, x+3= 0
odd function
graph is symmetrical with respect to the origin; f(-x)= -f(x)
even function
graph is symmetrical with respect to the y-axis; f(x) = f(-x)
one to one function
is when the horizontal line crosses the line graph 1 time and only -a parabola fails the horizontal line test so it is NOT a 1-1 function
midpoint formula
the average of x's and the average of y's = (x1+x2/2, y1+y2/2) = (x,y)
absolute values
the distance number is from zero; anything in bracket can be + or - |x| = x AND |-x| is also = x ie. |3x-6|-9 = -3, |3x-6|= -3+9, this means that 3x-6 =6 AND 3x-6 =-6
discriminant
the expression under the radical sign in the quadratic formula; b²-4ac from ax² + bx + c *used to determine the number and type of solutions to a quadratic equation =0 means 1 solution >0 means 2 solutions <0 means NO real solutions *solutions/roots: is the number of places where the parabola touches the x-axis
vertex
the highest or lowest point on the graph, usually a parabola; the expression outside the radical sign in the quadratic formula; x= -b/2a, y= ax² + bx + c
logarithmic function
the inverse of an exponential function; -if a^y = x, then log_a x = y and vice versa -log of x + log of y = log of x*y -log of x - log of y = log of x/y -log_e of x= ln of x -10^(log of x) = x ie 10^log2 = 2 *if there's 2 logs or 2 lns on both side, then they cancel out ie. log (x/4) = 2 a= 10, x= (x/4), y=2 a^y=x -> 10^2=x/4 -> x= 400
y intercept
the point at which the line crosses the y-axis (0,y) aka the y when x =0
x intercept
the x-coordinate of a point where a graph crosses the x-axis aka when y = 0
Absolute Value Inequalities
this type of inequalities usually only has 1 equation but produces 2 answers such that x(or the number after the sign) can be + or - * when you switch the sign on it, you must also flip the sign ie. >x becomes <-x ie. which is the solution to inequality: 4x+7 < -x ? the solutions will be 4x+7 <-x and 4x+7 >x
decimals
when multiplying/dividing these, it's sometimes better to turn them into exponents to get the whole numbers ie. 3.77 x2.8= 377 x10^-2 (28 x10^-1) = 10556 x( 10^-2-1)= 10556 x 10^-3 = 10.556
proper and improper fractions
when num>dom ie 5/4 and is >1 and when num<dom ie 4/5 <1
elimination method
when you add OR subtract two equations to eliminate one of the variables ie. given that 3x-8y=4 and 4.5x-6y=-6, what's the value of x? *need to multiply 1 or both equations to get 1 variable to have the same number and remove it from equations (narrowing down to 1 variable)
substitution method
when you have more than 1 equation involving 2+ variables, you can substitute 1 equation into another to narrow down 1 variable to solve it, then plug back in to find the other variable if needed. *can be a word question, then feel free to assign variables
square functions
y=x^2 will be open up y=-x^2 will be open down y=x^2+ 2 will be up and above by 2 y=x^2- 2 will be up and below 2
slope of the line
1. If you have 2 points on the graph, use m = y2-y1/x2-x1 2. If you have x and y, use y=mx+b -perpendicular lines will have a negative reciprocal slope of that line
Multiplying Fractions
1. Multiply the numerators. 2. Multiply the denominators. *make sure to cross-cancel any common numbers first to simplify the process
distance formula
1. use this when given 2 points on graph Square Root of ( x₂ - x₁)² + (y₂ - y₁)² 2. use the numbers of special triangles: 3:4:5, 5:12:13, 1:1:√2, 1: √3:2
exponent rules
-multiplying means addition -dividing means subtracting -common exponent can be grouped tgt ie x^b. y^b = (xy)^b -exponent of the exponent will be multiplied tgt ie 10^16^-4 = 10^ (16x-4) = 10^-64 *extra: roots can be turned into exponents and vice versa such that square root = x^1/2, cube root = x^1/3
common denominator
A denominator that is the same in two or more fractions. need to find the least common denominator when you have fractions with differing denominators by multiplying each fraction by 1 ie. 2/2, 5/5, 9/9
rational number
A number that can be written as a fraction
percentage
A ratio that compares a number to 100 1/2 = 50%. 1/4 = 25% 1/4 = 25% 1/5 = 20% 1/8 = 12.5% 1/25 = 4% 1/20 = 5% % =part/whole x100 % =is/of x100 *if it's not an easy %, just break up the whole numbers into 50%, 25%, 10% or 1% and add up or subtract to find the part
composite number
A whole number greater than 1 that has more than two factors aka the product of other whole numbers ie 6 is a composite bc divisible by 2 and 3
prime number
A whole number that has exactly two factors, 1 and itself aka only divisible by 1 and itself ie. factors of 51 -> 1 and 51 -> 3 and 17 therefore not a prime
inequalities
Algebraic statements that have ≠, <, >, ≤, or ≥ as their symbols of comparison ie. which of the following is a solution to this system of inequalities? 2x-3<x and -3x+8≤5 1. start by isolating x 2. combine both equations to find the range of x if needed 2x-x < 3, x<3 -3x≤ 5-8, x≤-3/-3 ≤1 therefore 1≤x<3
integer
All whole numbers (both positive and negative) and zero.
PROPORTIONS
An equation stating that two ratios are equal -basically 2 ratios equal each other which allows you to size up or down and manipulate the variables ie. a cake recipe for 10 ppl uses 2 eggs. How many eggs are needed for 15 ppl? 10ppl/2eggs = 15ppl/x eggs , cross multiply-> 10x = 30, x = 3 eggs
