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feasible region : the solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions of each inequality intersect | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
Gaussian elimination : using elementary row operations to obtain a matrix in row-echelon form | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
identity matrix : a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
inconsistent system : a system of linear equations with no common solution because they represent parallel lines, which have no point or line in common | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
independent system : a system of linear equations with exactly one solution pair(x,y)(x,y) | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
main diagonal : entries from the upper left corner diagonally to the lower right corner of a square matrix | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
matrix : a rectangular array of numbers | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
multiplicative inverse of a matrix : a matrix that, when multiplied by the original, equals the identity matrix | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
nonlinear inequality : an inequality containing a nonlinear expression | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
partial fraction decomposition : the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
partial fractions : the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
profit function : the profit function is written asP(x)=R(x)âC(x),P(x)=R(x)âC(x),revenue minus cost | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
revenue function : the function that is used to calculate revenue, simply written asR=xp,R=xp,wherex=x=quantity andp=p=price | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
row : a set of numbers aligned horizontally in a matrix | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
row operations : adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
row-echelon form : after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
row-equivalent : two matricesAAandBBare row-equivalent if one can be obtained from the other by performing basic row operations | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
scalar multiple : an entry of a matrix that has been multiplied by a scalar | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
solution set : the set of all ordered pairs or triples that satisfy all equations in a system of equations | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
substitution method : an algebraic technique used to solve systems of linear equations in which one of the two equations is solved for one variable and then substituted into the second equation to solve for the second variable | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
system of linear equations : a set of two or more equations in two or more variables that must be considered simultaneously. | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
system of nonlinear equations : a system of equations containing at least one equation that is of degree larger than one | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
system of nonlinear inequalities : a system of two or more inequalities in two or more variables containing at least one inequality that is not linear | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
An ellipse is 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. Each fixed point is called a focus (plural: foci). | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. SeeExample 1andExample 2. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. SeeExample 3andExample 4. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. SeeExample 5andExample 6. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
A hyperbola is 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/college-algebra-2e/pages/8-key-concepts |
The standard form of a hyperbola can be used to locate its vertices and foci. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. SeeExample 2andExample 3. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. SeeExample 4andExample 5. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
A parabola is 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/college-algebra-2e/pages/8-key-concepts |
The standard form of a parabola with vertex(0,0)(0,0)and thex-axis as its axis of symmetry can be used to graph the parabola. Ifp>0,p>0,the parabola opens right. Ifp<0,p<0,the parabola opens left. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
The standard form of a parabola with vertex(0,0)(0,0)and they-axis as its axis of symmetry can be used to graph the parabola. Ifp>0,p>0,the parabola opens up. Ifp<0,p<0,the parabola opens down. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
When given the focus and directrix of a parabola, we can write its equation in standard form. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
The standard form of a parabola with vertex(h,k)(h,k)and axis of symmetry parallel to thex-axis can be used to graph the parabola. Ifp>0,p>0,the parabola opens right. Ifp<0,p<0,the parabola opens left. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
The standard form of a parabola with vertex(h,k)(h,k)and axis of symmetry parallel to they-axis can be used to graph the parabola. Ifp>0,p>0,the parabola opens up. Ifp<0,p<0,the parabola opens down. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
A nondegenerate conic section has the general formAx2+Bxy+Cy2+Dx+Ey+F=0Ax2+Bxy+Cy2+Dx+Ey+F=0whereA,BA,BandCCare not all zero. The values ofA,B,A,B,andCCdetermine the type of conic. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
Equations of conic sections with anxyxyterm have been rotated about the origin. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
The general form can be transformed into an equation in thexâ²xâ²andyâ²yâ²coordinate system without thexâ²yâ²xâ²yâ²term. SeeExample 3andExample 4. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focusP(r,θ)P(r,θ)at the pole, and a line, the directrix, which is perpendicular to the polar axis. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
A conic is the set of all pointse=PFPD,e=PFPD,where eccentricityeeis a positive real number. Each conic may be written in terms of its polar equation. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
The polar equations of conics can be graphed. SeeExample 2,Example 3, andExample 4. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
Conics can be defined in terms of a focus, a directrix, and eccentricity. SeeExample 5andExample 6. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
We can use the identitiesr=x2+y2,x=rcosθ,r=x2+y2,x=rcosθ,andy=rsinθy=rsinθto convert the equation for a conic from polar to rectangular form. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/8-key-concepts |
x 2 a 2 + y 2 b 2 = 1 , a > b x 2 a 2 + y 2 b 2 = 1 , a > b | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
x 2 b 2 + y 2 a 2 = 1 , a > b x 2 b 2 + y 2 a 2 = 1 , a > b | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
( x â h ) 2 a 2 + ( y â k ) 2 b 2 = 1 , a > b ( x â h ) 2 a 2 + ( y â k ) 2 b 2 = 1 , a > b | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
( x â h ) 2 b 2 + ( y â k ) 2 a 2 = 1 , a > b ( x â h ) 2 b 2 + ( y â k ) 2 a 2 = 1 , a > b | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
x 2 a 2 â y 2 b 2 = 1 x 2 a 2 â y 2 b 2 = 1 | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
y 2 a 2 â x 2 b 2 = 1 y 2 a 2 â x 2 b 2 = 1 | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
( x â h ) 2 a 2 â ( y â k ) 2 b 2 = 1 ( x â h ) 2 a 2 â ( y â k ) 2 b 2 = 1 | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
( y â k ) 2 a 2 â ( x â h ) 2 b 2 = 1 ( y â k ) 2 a 2 â ( x â h ) 2 b 2 = 1 | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
y 2 = 4 p x y 2 = 4 p x | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
x 2 = 4 p y x 2 = 4 p y | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
( y â k ) 2 = 4 p ( x â h ) ( y â k ) 2 = 4 p ( x â h ) | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
( x â h ) 2 = 4 p ( y â k ) ( x â h ) 2 = 4 p ( y â k ) | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
A x 2 + B x y + C y 2 + D x + E y + F = 0 A x 2 + B x y + C y 2 + D x + E y + F = 0 | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
x = x â² cos θ â y â² sin θ y = x â² sin θ + y â² cos θ x = x â² cos θ â y â² sin θ y = x â² sin θ + y â² cos θ | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
θ , where cot ( 2 θ ) = A â C B θ , where cot ( 2 θ ) = A â C B | https://openstax.org/books/college-algebra-2e/pages/8-key-equations |
angle of rotation : an acute angle formed by a set of axes rotated from the Cartesian plane where, ifcot(2θ)>0,cot(2θ)>0,thenθθis between(0°,45°);(0°,45°);ifcot(2θ)<0,cot(2θ)<0,thenθθis between(45°,90°);(45°,90°);and ifcot(2θ)=0,cot(2θ)=0,thenθ=45°θ=45° | https://openstax.org/books/college-algebra-2e/pages/8-key-terms |
center of a hyperbola : the midpoint of both the transverse and conjugate axes of a hyperbola | https://openstax.org/books/college-algebra-2e/pages/8-key-terms |
center of an ellipse : the midpoint of both the major and minor axes | https://openstax.org/books/college-algebra-2e/pages/8-key-terms |
conic section : any shape resulting from the intersection of a right circular cone with a plane | https://openstax.org/books/college-algebra-2e/pages/8-key-terms |
conjugate axis : the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints | https://openstax.org/books/college-algebra-2e/pages/8-key-terms |
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/college-algebra-2e/pages/8-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/college-algebra-2e/pages/8-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/college-algebra-2e/pages/8-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/college-algebra-2e/pages/8-key-terms |
foci : plural of focus | https://openstax.org/books/college-algebra-2e/pages/8-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/college-algebra-2e/pages/8-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/college-algebra-2e/pages/8-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/college-algebra-2e/pages/8-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/college-algebra-2e/pages/8-key-terms |
major axis : the longer of the two axes of an ellipse | https://openstax.org/books/college-algebra-2e/pages/8-key-terms |
minor axis : the shorter of the two axes of an ellipse | https://openstax.org/books/college-algebra-2e/pages/8-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/college-algebra-2e/pages/8-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/college-algebra-2e/pages/8-key-terms |
polar equation : an equation of a curve in polar coordinatesrrandθθ | https://openstax.org/books/college-algebra-2e/pages/8-key-terms |
transverse axis : the axis of a hyperbola that includes the foci and has the vertices as its endpoints | https://openstax.org/books/college-algebra-2e/pages/8-key-terms |
A sequence is a list of numbers, called terms, written in a specific order. | https://openstax.org/books/college-algebra-2e/pages/9-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/college-algebra-2e/pages/9-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/college-algebra-2e/pages/9-key-concepts |
Recursive formulas define each term of a sequence using previous terms. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
Recursive formulas must state the initial term, or terms, of a sequence. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
A set of terms can be written by using a recursive formula. SeeExample 5andExample 6. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
A factorial is a mathematical operation that can be defined recursively. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The factorial ofnnis the product of all integers from 1 tonnSeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant. | https://openstax.org/books/college-algebra-2e/pages/9-key-concepts |
The constant between two consecutive terms is called the common difference. | https://openstax.org/books/college-algebra-2e/pages/9-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/college-algebra-2e/pages/9-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/college-algebra-2e/pages/9-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/college-algebra-2e/pages/9-key-concepts |
As with any recursive formula, the initial term of the sequence must be given. | https://openstax.org/books/college-algebra-2e/pages/9-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/college-algebra-2e/pages/9-key-concepts |
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