Khan Academy Algebra
example of an open interval
(-2,3) or x is greater than negative 2 and is less than 3
what is the slope of a horizontal line?
0
example of sequence
1,3,5,7,9 difference: +2
tangent line
A line that intersects a curve once and only once
relative maximum
A point on the graph of a function where no other nearby points have a greater y-coordinate.
relative minimum
A point on the graph of a function where no other nearby points have a lesser y-coordinate.
negative function
A portion of a function where the y-values are less than 0 for all x-values
what is a unit rate
A rate with a denominator of 1.
arithmetic sequence
A sequence in which the difference between any two consecutive terms is the same.
example of a relationship in a function
A tree grows 20 centimeters per year. so the height is related to its age using the function h. h(y)=20y
what is an equation
An Equation is a statement that two mathematical expressions are equal.
closed interval
An interval that includes both of its ending points and all the points between these ending points
standard form
Ax+By=C where A, B, and C are real numbers and A & B are not both zero.
complex number symbol
C
Equation vs Expression
Equation relation sign (=or >) you solve is a sentence DO SAME TO BOTH SIDES of the equal sign (for these allowable operations: + − × ÷ √ ( )ⁿ add, subtract, multiply, divide, "take a root" and "raise to a power") Expression no relation size you simplify (not solve) ... can multiply by 1 or combine like terms and factor is a phrase
Imaginary number symbol
I
example of independent and dependent variables
John works x hours and makes $3 per hour. what is his total in y after x hours? 3x=y the dependent is total because it depends on how many hours worked the independent is y because it is the variable being manipulated
natural numbers symbol
N
rational numbers symbol
Q
Real numbers symbol
R
what is function notation?
Replacing the y with f(x) Example: y = 2x + 3 to f(x) = 2x + 3
what is a function?
System-defined formula
Double inequality
Two inequalities can often be combined into a DOUBLE INEQUALITY or COMPOUND INEQUALITY to indicate that the numbers lie between two fixed values. e.g. -16<3x+5<20
compound inequality
Two or more inequalities joined together by "and" or "or"
what is the slope of a vertical line?
Undefined, because we can't divide by 0.
integers symbol
Z
example of an opened and closed interval
[-2,3) or x is greater than or equal to negative 2 and is less than 3
example of a closed interval
[-2,3] or x is greater than or equal to -2 and x is less than or equal to 3
set builder notation
a concise way of writing a solution set
positive function
a portion of a function where the y-values are greater than 0 for all x-values
geometric sequence
a sequence in which each term is found by multiplying the previous term by the same number
recursive formula of arithmetic sequence
a(n)= a(n-1)+difference a= output n= nth term
explicit formula of arithmetic sequence
a(n)=a1+d(n-1) n= nth term a=function a in terms of term n
example of variable
a+1=x a and x are variables
arithmetic sequences can use only what two operations?
addition or subtraction
what is a linear equation?
algebraic expression where the highest power is 0 i.e. 2x+2
what is an interval
all numbers between two numbers
domain of sequence
all possible inputs of function since domain is in terms of nth term its always positive
algebraic equation
an expression with one or more variables i.e. 2x+1=5
open interval
an interval that does not include its endpoints
what is an open and closed interval?
an interval where one endpoint is included and the other is not
what is an algebraic expression
an set of terms that represent a value but no equal sign
what do compound inequalities joined by "and" mean
both inequalities must be satisfied for the solution to be considered true
why is standard form useful?
can be used to find intercepts of a line by setting x or y equal to zero
slope of a line
change in y over change in x. a.k.a rise over run
what can you do to terms with the same variable in an equation?
combine them i.e. 2x+3y-2x 1x+3y
common difference of an arithmetic sequence
constant difference between consecutive terms
how do you set up unit rate?
dependent variable/independent variable i.e. distance per hour depends on speed 45m/hr i.e. 45m/1hr
what should you do to parenthesis before combining like terms
distribute the parenthesis properly i.e 2x(3x+25)-2x 6x^2+48x
inverse of multiplication
division a(1/a)=1
how do you get rid of a variable on one side of the equation?
do the opposite operation on both sides that cancels it out on the desired side
what do compound inequalities joined by "or" mean
either of the inequalities can be true to considered true
what is a true equation?
expression on both sides of equal sign are equal
what is an equivalent expression?
expressions that represent the same value i.e. 4a+8=2(2a+4)
example of interval domain
f(x)=3/x x= any real value except zero (3/0= undefined) (-infinity,0) U (0, infinity)
common example of functions
f(x)=x^2 where x is the input, f(x) is the output, ^2 is the relationship
plotting greater than or equal to or less than or equal to on number line
filled in circle + line
when multiplying or dividing inequalities by a negative what happens to the inequality sign?
flip it i.e. -2x>4 x<-2
interval of domain
formal way of describing continuous set of real values that can describe a domain
absolute maximum of a function
highest value of a function written as (x,y)
what is the relationship of a function?
how the input relates to the output
when does an algebraic equation have only one solution?
if there is one number equaling a variable i.e. x=5
visualize a function
input, function, output
what are the three parts of a function?
input, relationship, output
absolute minimum of a function
lowest value of a function
what is the input of a function?
number that is inserted into the function f(X)=a
plotting greater than or less than on number line
open circle + line
common ratio of geometric sequence
ratio between consecutive terms represented as a multiplier
how does a function work?
relates an input to an output
What is a a way to represent an interval in a fancy way
set builder notation
What is the "domain" of a function?
set of all inputs in which the function has a defined output
Sum of Consecutive odd or even Integers
set up as x, x+2, x+4 ( represents relations ship between x and next odd or even integer e.g 1,3,5)
inverse of addition
subtraction a+(-a)=0
what is a variable
symbol for an unknown value
what does the slope-intercept form represent
the equation of a line
secant line
the line through two points on a curve
what is the output of a function?
the result of the input
what is the range of a function?
the set of all possible outputs
What is the function name of a function
the variable before the parentheses i.e. F(x)
what is the argument of a function?
the variable inside the parentheses of a function i.e. f(X)
x-intercept
the x-coordinate of a point where a graph crosses the x-axis
Y-intercept
the y-coordinate of a point where a graph crosses the y-axis
what must be true of the input values of a function
they can only represent one value of the output.
what happens when you divide by 0?
undefined
average rate of change of a curve
use coordinates of secant line to find average rate of change over an interval
what is an independent variable?
variable that is manipulated
what is a dependent variable?
variable that is measured
when does an equation have infinite solutions?
when both sides of the equation have equal numerical values i.e. 13=13
when does an equation have zero solutions
when there are two different values on each side of the equal sign i.e. 4=6 (not true)
decreasing function interval
when y<0
increasing function interval
when y>0
the independent variable is usually on what axis?
x-axis
equation for a vertical line
x=a
what is point-slope form?
y minus y1 equals m times x minus x1
The dependent variable is usually on what axis?
y-axis
point-slope form
y-y1=m(x-x1)
slope formula given two points
y2-y1/x2-x1
equation for a horizontal line
y=b
slope intercept form
y=mx+b, where m is the slope and b is the y-intercept of the line.
set builder notation example
{ k member of reals | k > 5 }