text stringlengths 2 2.33k | source stringclasses 826
values |
|---|---|
degenerate conic sections : any of the possible shapes formed when a plane intersects a double cone through the apex. Types of degenerate conic sections include a point, a line, and intersecting lines. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
directrix : a line perpendicular to the axis of symmetry of a parabola; a line such that the ratio of the distance between the points on the conic and the focus to the distance to the directrix is constant | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
eccentricity : the ratio of the distances from a pointPPon the graph to the focusFFand to the directrixDDrepresented bye=PFPD,e=PFPD,whereeeis a positive real number | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
ellipse : the set of all points(x,y)(x,y)in a plane such that the sum of their distances from two fixed points is a constant | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
foci : plural of focus | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
focus (of a parabola) : a fixed point in the interior of a parabola that lies on the axis of symmetry | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
focus (of an ellipse) : one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point(x,y)(x,y)on the ellipse is a constant | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
hyperbola : the set of all points(x,y)(x,y)in a plane such that the difference of the distances between(x,y)(x,y)and the foci is a positive constant | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
latus rectum : the line segment that passes through the focus of a parabola parallel to the directrix, with endpoints on the parabola | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
major axis : the longer of the two axes of an ellipse | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
minor axis : the shorter of the two axes of an ellipse | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
nondegenerate conic section : a shape formed by the intersection of a plane with a double right cone such that the plane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
parabola : the set of all points(x,y)(x,y)in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
polar equation : an equation of a curve in polar coordinatesrrandθθ | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
transverse axis : the axis of a hyperbola that includes the foci and has the vertices as its endpoints | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
A sequence is a list of numbers, called terms, written in a specific order. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
Explicit formulas define each term of a sequence using the position of the term. SeeExample 1,Example 2, andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
An explicit formula for thenthnthterm of a sequence can be written by analyzing the pattern of several terms. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
Recursive formulas define each term of a sequence using previous terms. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
Recursive formulas must state the initial term, or terms, of a sequence. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
A set of terms can be written by using a recursive formula. SeeExample 5andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
A factorial is a mathematical operation that can be defined recursively. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The factorial ofnnis the product of all integers from 1 tonnSeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The constant between two consecutive terms is called the common difference. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. SeeExample 2andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
A recursive formula for an arithmetic sequence with common differenceddis given byan=anâ1+d,nâ¥2.an=anâ1+d,nâ¥2.SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
As with any recursive formula, the initial term of the sequence must be given. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
An explicit formula for an arithmetic sequence with common differenceddis given byan=a1+d(nâ1).an=a1+d(nâ1).SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
An explicit formula can be used to find the number of terms in a sequence. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
In application problems, we sometimes alter the explicit formula slightly toan=a0+dn.an=a0+dn.SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The constant ratio between two consecutive terms is called the common ratio. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The common ratio can be found by dividing any term in the sequence by the previous term. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. SeeExample 2andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
A recursive formula for a geometric sequence with common ratiorris given byan=ranâ1an=ranâ1fornâ¥2nâ¥2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
As with any recursive formula, the initial term of the sequence must be given. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
An explicit formula for a geometric sequence with common ratiorris given byan=a1rnâ1.an=a1rnâ1.SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
In application problems, we sometimes alter the explicit formula slightly toan=a0rn.an=a0rn.SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The sum of the terms in a sequence is called a series. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The sum of the terms in an arithmetic sequence is called an arithmetic series. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The sum of the firstnnterms of an arithmetic series can be found using a formula. SeeExample 2andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The sum of the terms in a geometric sequence is called a geometric series. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The sum of the firstnnterms of a geometric series can be found using a formula. SeeExample 4andExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The sum of an infinite series exists if the series is geometric withâ1<r<1.â1<r<1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
If the sum of an infinite series exists, it can be found using a formula. SeeExample 6,Example 7, andExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
If one event can occur inmmways and a second event with no common outcomes can occur innnways, then the first or second event can occur inm+nm+nways. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
If one event can occur inmmways and a second event can occur innnways after the first event has occurred, then the two events can occur inmÃnmÃnways. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
A permutation is an ordering ofnnobjects. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
If we have a set ofnnobjects and we want to chooserrobjects from the set in order, we writeP(n,r).P(n,r). | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
Permutation problems can be solved using the Multiplication Principle or the formula forP(n,r).P(n,r).SeeExample 3andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
A selection of objects where the order does not matter is a combination. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
Givennndistinct objects, the number of ways to selectrrobjects from the set isC(n,r)C(n,r)and can be found using a formula. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
A set containingnndistinct objects has2n2nsubsets. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
(nr)(nr)is called a binomial coefficient and is equal toC(n,r).C(n,r).SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The Binomial Theorem allows us to expand binomials without multiplying. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
We can find a given term of a binomial expansion without fully expanding the binomial. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The probabilities in a probability model must sum to 1. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
The probability of the complement of an event is the difference between 1 and the probability that the event occurs. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-concepts |
0 ! = 1 1 ! = 1 n ! = n ( n â 1 ) ( n â 2 ) ⯠( 2 ) ( 1 ) , for n ⥠2 0 ! = 1 1 ! = 1 n ! = n ( n â 1 ) ( n â 2 ) ⯠( 2 ) ( 1 ) , for n ⥠2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
a n = a n â 1 + d , n ⥠2 a n = a n â 1 + d , n ⥠2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
a n = a 1 + d ( n â 1 ) a n = a 1 + d ( n â 1 ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
a n = r a n â 1 , n ⥠2 a n = r a n â 1 , n ⥠2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
a n = a 1 r n â 1 a n = a 1 r n â 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
S n = n ( a 1 + a n ) 2 S n = n ( a 1 + a n ) 2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
S n = a 1 ( 1 â r n ) 1 â r , r â 1 S n = a 1 ( 1 â r n ) 1 â r , r â 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
S n = a 1 1 â r , r â 1 S n = a 1 1 â r , r â 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
P ( n , r ) = n ! ( n â r ) ! P ( n , r ) = n ! ( n â r ) ! | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
C ( n , r ) = n ! r ! ( n â r ) ! C ( n , r ) = n ! r ! ( n â r ) ! | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
( x + y ) n = â k â 0 n ( n k ) x n â k y k ( x + y ) n = â k â 0 n ( n k ) x n â k y k | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
( r + 1 ) t h ( r + 1 ) t h term of a binomial expansion | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
P ( E ) = n ( E ) n ( S ) P ( E ) = n ( E ) n ( S ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
P ( E ⪠F ) = P ( E ) + P ( F ) â P ( E â© F ) P ( E ⪠F ) = P ( E ) + P ( F ) â P ( E â© F ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
P ( E ⪠F ) = P ( E ) + P ( F ) P ( E ⪠F ) = P ( E ) + P ( F ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
P ( E ' ) = 1 â P ( E ) P ( E ' ) = 1 â P ( E ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-equations |
Addition Principle : if one event can occur inmmways and a second event with no common outcomes can occur innnways, then the first or second event can occur inm+nm+nways | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
annuity : an investment in which the purchaser makes a sequence of periodic, equal payments | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
arithmetic sequence : a sequence in which the difference between any two consecutive terms is a constant | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
arithmetic series : the sum of the terms in an arithmetic sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
binomial coefficient : the number of ways to chooserobjects fromnobjects where order does not matter; equivalent toC(n,r),C(n,r),denoted(nr)(nr) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
binomial expansion : the result of expanding(x+y)n(x+y)nby multiplying | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
Binomial Theorem : a formula that can be used to expand any binomial | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
combination : a selection of objects in which order does not matter | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
common difference : the difference between any two consecutive terms in an arithmetic sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
common ratio : the ratio between any two consecutive terms in a geometric sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
complement of an event : the set of outcomes in the sample space that are not in the eventEE | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
diverge : a series is said to diverge if the sum is not a real number | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
event : any subset of a sample space | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
experiment : an activity with an observable result | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
explicit formula : a formula that defines each term of a sequence in terms of its position in the sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
finite sequence : a function whose domain consists of a finite subset of the positive integers{1,2,â¦n}{1,2,â¦n}for some positive integernn | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.