alAlgebra and Trigonometry Notes (Combinatorics too)
Rational number – decimal notation either repeats or terminates. Whole numbers
Irrational Number – where no decimal repeats
Interval notation – (a,b) pints a,b are endpoints of interval. Some intervals bound only one way. [a,
infinity)
Absolute value – (a – b) or (b – a)
Commutative properties of + and *: (a+b)=(b+a) (ab=ba)
Associative properties of + and *: a + (b+c) = (a+b) + c a(bc)=(ab)c
Additive identity property: a + 0 = 0+ a = a
Additive inverse property: -a + a = a + (-a) = 0
Multiplicative Identity property: a * 1 = 1 * a = a
Multiplicative Inverse property: a * 1/a = 1/a * a = a
𝑎 −𝑚 𝑏𝑛
Integers as exponents =
𝑏−𝑛 𝑎𝑚
Adding and Subtracting polynomials – collect like terms
Multiplying – coefficients then variables.
Chapter One
Graph – solutions to an equation
Find solutions and plot data
Distance formula
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Midpoint formula
𝑥 +𝑥
( 1 2 2 ) same for y coordinates
Equation of a circle
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
1.2
functions – correspondence between domain and range
Any number of domains to one range
Relation – correspondence between domain and any number of ranges
So, for ordered pairs, functions can’t have two coordinate pairs with same first coordinate
F(x) means f of x, for every x input there is a y output
1.3
F(x) = mx + b
Constant function f(x) = b
Identity function f(x) = x
Horizontal line f(x) = b
Vertical Line f(x) = a (not a function)
For horizontal lines – change in y is zero and change in x is nonzero
Vertical lines – change in y is nonzero and change in x is zero. Slope undefined, cannot divide by zero
, Average rate of change – find the change between two data points on a graph
1.4,1.5,1.6
Point slope equation passing through (𝑥1 , 𝑦1 )
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Find slope and then find plug in coordinates to make equation of a line
Parallel lines must have same slope
Two lines with slopes are perpendicular if the product of their slopes is -1
For perpendicular – take reciprocal and negative
Cure fit to find a function that fits – or regression the data!!!
Equations with same solution set are equivalent equations
Use addition and multiplication principles for equation-solving
Simple interest formula – I = P*r*t
Liner inequalities take same principles as equation-solving
Negatives flip the inequality sign
Then put into interval notation or domain and range notation
For compound inequalities, put into con junction
When absolute is on a graph, inside absolute you move position left or right
2.1
Max and min
Increase / decrease
Greatest Integer Function: greatest integer less than or equal to [lxl]
Round down to lowest integer
For piecewise functions add together domains of both and plot
2.2
The set of numbers on the domain of a function (𝑓 + 𝑔) are the set of numbers in both domains.
𝑓
For (𝑔)(𝑥) g(x) can’t equal zero. when polynomials make a rational expression and do rational
expression things with it
For difference quotient, sub f(x) into place. Slope of the line is the secant line
For 𝑓(𝑥 + ℎ) , the (𝑥 + ℎ) goes in all the x values of 𝑓(𝑥)
Remember complex rational expressions
2.3
For 𝑓(𝑔(𝑥)) the 𝑔(𝑥)input goes into every 𝑓(𝑥) variable
For domain use common sense. Don’t let roots be negative or denominators be zero
To decompose, find components that will compose a function. Really think.
2.4
Algebraic Tests of Symmetry
x-axis: if replacing y with -y produces and = equation then it is symmetric to the x-axis
y-axis: if replacing x with -x produces and = equation then it is symmetric to the y-axis
origin: if replacing x with -x and y with -y produces = equation then it is symmetric to the origin
Even & Odd Functions - Transformations
If graph of function f is symmetric to y-axis it is even function. 𝑓(𝑥) = 𝑓(−𝑥)
If graph of function f is symmetric to origin it is odd function. 𝑓(−𝑥) = −𝑓(𝑥)
Rational number – decimal notation either repeats or terminates. Whole numbers
Irrational Number – where no decimal repeats
Interval notation – (a,b) pints a,b are endpoints of interval. Some intervals bound only one way. [a,
infinity)
Absolute value – (a – b) or (b – a)
Commutative properties of + and *: (a+b)=(b+a) (ab=ba)
Associative properties of + and *: a + (b+c) = (a+b) + c a(bc)=(ab)c
Additive identity property: a + 0 = 0+ a = a
Additive inverse property: -a + a = a + (-a) = 0
Multiplicative Identity property: a * 1 = 1 * a = a
Multiplicative Inverse property: a * 1/a = 1/a * a = a
𝑎 −𝑚 𝑏𝑛
Integers as exponents =
𝑏−𝑛 𝑎𝑚
Adding and Subtracting polynomials – collect like terms
Multiplying – coefficients then variables.
Chapter One
Graph – solutions to an equation
Find solutions and plot data
Distance formula
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Midpoint formula
𝑥 +𝑥
( 1 2 2 ) same for y coordinates
Equation of a circle
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
1.2
functions – correspondence between domain and range
Any number of domains to one range
Relation – correspondence between domain and any number of ranges
So, for ordered pairs, functions can’t have two coordinate pairs with same first coordinate
F(x) means f of x, for every x input there is a y output
1.3
F(x) = mx + b
Constant function f(x) = b
Identity function f(x) = x
Horizontal line f(x) = b
Vertical Line f(x) = a (not a function)
For horizontal lines – change in y is zero and change in x is nonzero
Vertical lines – change in y is nonzero and change in x is zero. Slope undefined, cannot divide by zero
, Average rate of change – find the change between two data points on a graph
1.4,1.5,1.6
Point slope equation passing through (𝑥1 , 𝑦1 )
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Find slope and then find plug in coordinates to make equation of a line
Parallel lines must have same slope
Two lines with slopes are perpendicular if the product of their slopes is -1
For perpendicular – take reciprocal and negative
Cure fit to find a function that fits – or regression the data!!!
Equations with same solution set are equivalent equations
Use addition and multiplication principles for equation-solving
Simple interest formula – I = P*r*t
Liner inequalities take same principles as equation-solving
Negatives flip the inequality sign
Then put into interval notation or domain and range notation
For compound inequalities, put into con junction
When absolute is on a graph, inside absolute you move position left or right
2.1
Max and min
Increase / decrease
Greatest Integer Function: greatest integer less than or equal to [lxl]
Round down to lowest integer
For piecewise functions add together domains of both and plot
2.2
The set of numbers on the domain of a function (𝑓 + 𝑔) are the set of numbers in both domains.
𝑓
For (𝑔)(𝑥) g(x) can’t equal zero. when polynomials make a rational expression and do rational
expression things with it
For difference quotient, sub f(x) into place. Slope of the line is the secant line
For 𝑓(𝑥 + ℎ) , the (𝑥 + ℎ) goes in all the x values of 𝑓(𝑥)
Remember complex rational expressions
2.3
For 𝑓(𝑔(𝑥)) the 𝑔(𝑥)input goes into every 𝑓(𝑥) variable
For domain use common sense. Don’t let roots be negative or denominators be zero
To decompose, find components that will compose a function. Really think.
2.4
Algebraic Tests of Symmetry
x-axis: if replacing y with -y produces and = equation then it is symmetric to the x-axis
y-axis: if replacing x with -x produces and = equation then it is symmetric to the y-axis
origin: if replacing x with -x and y with -y produces = equation then it is symmetric to the origin
Even & Odd Functions - Transformations
If graph of function f is symmetric to y-axis it is even function. 𝑓(𝑥) = 𝑓(−𝑥)
If graph of function f is symmetric to origin it is odd function. 𝑓(−𝑥) = −𝑓(𝑥)
