15.1 词汇表(A-E)

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  A
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  abscissa  

  The $\begin{align*}x\end{align*}$ -coordinate of the ordered pair that represents a plotted point on a Cartesian plane. For the point (3, 7), 3 is the abscissa.

  absolute maximum/minimum

  The highest and lowest points of a function, respectively. When referring to the absolute maximum/minimum value, use the $\begin{align*}y\end{align*}$   $\begin{align*}\left|ax+b\right| > c\end{align*}$  value. 

  absolute value

  The distance that a number is from zero. Absolute values are never negative.

  absolute value equation

  An equation that involves an absolute value expression.

  absolute value inequality

  Absolute value linear inequalities can have one of four forms:  $\begin{align*}\left|ax+b\right|> c\end{align*}$ , $\begin{align*}\left|ax+b\right|<c\end{align*}$ , $\begin{align*}\left|ax+b\right|\ge c\end{align*}$ , or  $\begin{align*}\left|ax+b\right| \le c\end{align*}$ .

  addition property of equality

  You can add the same quantity to both sides of an equation without changing the relative truth of the statement. If $\begin{align*}2x=6\end{align*}$ , then $\begin{align*}2x+2=6+2\end{align*}$ .

  addition property of inequality

  You can add a quantity to both sides of an inequality without changing the sense of the inequality. If $\begin{align*}x>3\end{align*}$ , then $\begin{align*}x+2>3+2\end{align*}$ .

  additive identity

  0 is the additive identity.

  additive inverse

  -a is the additive inverse of a.

  algebraic (variable) expression

  An expression that includes variables. 

  alternating sequence

  A sequence whose terms alternate between having a positive or a negative sign.

  arc

  A section of the circumference of a circle.

  area

  The space within the perimeter of a two-dimensional figure.

  arithmetic sequence

  An arithmetic sequence has a common difference between each two consecutive terms. It is also known as an arithmetic progression.

  associative property

  The order in which three or more values are grouped for multiplication or addition will not affect the product or sum. For example,  $\begin{align*}(a+b)+c=a+(b+c)\end{align*}$   and  $\begin{align*}(ab)c=a(bc)\end{align*}$ .

  asymptote

  A boundary line for the graph of a relation.

  axis (double right-circular cone)

  A central line to form the double right-circular cone.

  axis of symmetry

  In a parabola, the vertical line that passes through the vertex of the parabola. The parabola is symmetric about this line. 

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  base

  When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression $\begin{align*}{32}^{4}\end{align*}$ , $\begin{align*}32\end{align*}$  is the base, and $\begin{align*}4\end{align*}$  is the exponent.

  binomial  

  A polynomial with two terms. The prefix "bi" means "two."

  Binomial Theorem

  An efficient formula for calculating the expansion of binomials. It states that

  $\begin{align*}{x+ y}^{n} = \sum_{r = 0}^n \end{align*}$   $\begin{align*} \binom{n}{r} {x}^{n-r}{y}^{r}.\end{align*}$  
  

  boundary line

  A line that bounds at least one side of a graph.

  bounded interval

  An interval with two endpoints or one that does not continue forever in either direction.

  Bounds Theorem

  The Bounds Theorem states that c is an upper bound for the real zeros if, when you divide by x−c, the coefficients of the quotient and the remainder all have the same sign. c is a lower bound for the real zeros if, when you divide by x−c, the coefficients of the quotient and the remainder have alternating signs.

  branches

  The two curves of a hyperbola.

  breadth

  The distance between the halves of a function with two symmetric sides (such as an absolute value function).

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  cardinality

  The number of elements in a set.

  Cartesian coordinate system

  A grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.

  center

  A middle or central point.

  check

  The process of determining whether a solution is correct.

  chord

  A straight line across a circle, intersecting the circle in two places, but not passing through the circle's center.

  circle

  The set of all points at a specific distance from a given point in two dimensions.

  circumference

  The measure of the distance around the outside edge of a circle.

  closed interval

  An interval where both of the endpoints are included.

  co-vertices

  The endpoints of the minor axis of an ellipse.

  coefficient

  The number in front of a variable.

  coincide

  Coincide means the graphs have all points in common.

  combination

  Distinct arrangements of a specified number of objects without regard to order of selection from a specified set.

  combined variation

  A combination of direct, inverse, and joint variation.

  combining like terms

  If the same variable appears in an expression more than once, the terms with that variable can possibly  be combined by addition or subtraction. 

  common difference

  Every arithmetic sequence has a common or constant difference between consecutive terms. For example, in the sequence 5, 8, 11, 14..., the common difference is "3."

  common logarithm

  A log with base 10. The log is usually written without the base.

  common ratio

  Every geometric sequence has a common ratio, or a constant ratio between consecutive terms. For example, in the sequence 2, 6, 18, 54..., the common ratio is 3.

  commutative property

  Changing the order of the numbers in an addition or multiplication problem does not change the sum. For example,  $\begin{align*}2+3=3+2\end{align*}$   and $\begin{align*}2\times 3 = 3\times 2\end{align*}$ .

  complement

  All of the elements in the universal set, not in the set itself.

  completing the square

  A common method for rewriting quadratics. It refers to making a perfect square trinomial by adding the square of 1/2 of the coefficient of the $\begin{align*}x\end{align*}$ -term.

  complex conjugate

  Pairs of complex numbers of the form $\begin{align*}a+bi\end{align*}$  and $\begin{align*}a-bi\end{align*}$ . When complex conjugates are multiplied, the result is a single real number.

  Complex Conjugate Theorem

  Complex zeros or roots of a polynomial function occur in conjugate pairs; that is, if a+bi is a zero, so is a−bi its complex conjugate.

  complex fraction

  A fraction with rational expression(s) in the numerator and denominator (a fraction composed of other fractions).

  complex number

  The sum of a real number and an imaginary number, written in the form $\begin{align*}a+bi\end{align*}$ .

  compose

  Performing the operation composition.

  composite number

  A number that has more than two factors.

  composition

  The operation in which we input one function into another.

  compound

  The process of calculating interest in a compound interest situation.

  compound inequality

  An inequality that combines two other inequalities. For example,  $\begin{align*}5<x<10\end{align*}$ , meaning $\begin{align*}x>5\end{align*}$  and $\begin{align*}x<10\end{align*}$ .

  compound interest

  Interest earned on the total amount at the time it is compounded, including previously earned interest.

  compounding continuously

  An exponential growth model used to estimate compounding daily.

  cone

  A solid three-dimensional figure with a circular base and one vertex.

  confocal

  Two conic sections that share a common focus.

  conic sections

  Curves that can be created by the intersection of a double cone and a plane. They include circles, ellipses, parabolas, and hyperbolas.

  consistent system

  A system of equations is consistent if it has at least one solution.

  constant (numeric)

  A value that does not change. In algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable, such as $\begin{align*}x,y,\end{align*}$  or $\begin{align*}a\end{align*}$ .

  constant (function)

  A section of a function's graph in which the y-values stay the same; that is,  $\begin{align*}f({x}_{2})=f({x}_{1})\end{align*}$ .

  constant of proportionality

  The constant ratio of two proportional quantities, such as $\begin{align*}x\end{align*}$  and $\begin{align*}y\end{align*}$ . Also  known as the constant of variation, i t is com monly represented as $\begin{align*}k\end{align*}$ . 

  constant of variation

  The constant ratio of two proportional quantities, such as $\begin{align*}x\end{align*}$  and $\begin{align*}y\end{align*}$ . Also known as the constant of proportionality, it is  commonly represented as $\begin{align*}k.\end{align*}$  

  constant term

  The term that is just a number (or the $\begin{align*}{x}^{0}\end{align*}$ -term).

  constraint

  A limitation.

  continuous

  A function without breaks or gaps.

  converge

  If the series gets close to—or "approaches"—a particular number.

  conversion factor

  A ratio used to convert from one unit of measurement to another.

  coordinate

  One of the values in an ordered pair.

  coordinate pair

  A coordinate pair, $\begin{align*}(x,y)\end{align*}$ , describes the location of a point on a coordinate grid.

  coordinate plane

  A grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian plane.

  cross-multiplication

  If $\begin{align*}a,b,c,\end{align*}$  and $\begin{align*}d\end{align*}$  are real numbers, with $\begin{align*}b\neq 0\end{align*}$  and $\begin{align*}d\neq 0\end{align*}$  and if $\begin{align*}\frac{a}{b}= \frac{c}{d}\end{align*}$ , then $\begin{align*}ad=bc\end{align*}$ .

  cube root

  When the index of a radical is 3, the root is called the cube root. 

  cylinder

  A solid figure with two parallel congruent circular bases.

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  decay factor

  The fractional or decimal multiple by which an exponential function decreases. Decay factors are always less than 1. Populations, depreciated values, and radioactivity commonly use decay factors.

  decreasing

  A function is decreasing over an interval if its $\begin{align*}y\end{align*}$ -values are getting smaller over the interval. The graph will go down from left to right over the interval.

  degenerate conic section

  A conic that does not have the usual properties of a conic section. Since some of the coefficients of the general conic equation are zero, the basic shape of the conic is merely a point, a line, or a pair of intersecting lines.

  degree

  The degree of a polynomial is the largest exponent of the polynomial.

  dependent system

  A system of equations is dependent if two equations coincide.

  dependent variable

  The output variable in an equation or function, commonly represented by $\begin{align*}y\end{align*}$  or $\begin{align*}f(x)\end{align*}$ .

  Descartes's Rule of Signs

  The number of positive real zeros is the number of sign changes of f(x) or an even number less. The number of negative real zeros is the number of sign changes of f(-x) or an even number less.

  diameter

  The measure of the distance across the center of a circle. The diameter is equal to twice the measure of the radius.

  difference

  The result of a subtraction operation.

  difference of two squares

  A quadratic equation in the form $\begin{align*}{a}^{2} - {b}^{2}\end{align*}$ .

  difference of two cubes

  The difference of cubes formula is $\begin{align*}{a}^{3} - {b}^{3} = (a - b) ({a}^{2} + ab + {b}^{2})\end{align*}$ .

  dilation

  To reduce or enlarge a figure according to a scale factor.

  dimensional analysis

  A process for converting from one unit to another.

  direct variation

  When the dependent variable grows large or small as the independent variable does.

  directrix

  The line that the parabola seems to curve away from. All points on a parabola are equidistant from the focus of the parabola and the directrix of the parabola.

  discriminant

  The part of the quadratic formula under the radical  $\begin{align*}{b}^{2}-4ac\end{align*}$ . A positive discriminant suggests two real roots to the quadratic equation, a zero suggests one real root with multiplicity two, and a negative indicates two complex roots.

  disjoint sets

  When the intersection of two sets is empty.

  distance formula

  The distance between two points $\begin{align*}({x}_{1}, {y}_{1})\end{align*}$  and $\begin{align*}({x}_{2}, {y}_{2})\end{align*}$  can be defined as $\begin{align*}d=\sqrt{{({x}_{2} - {x}_{1})}^{2}+ {({y}_{2}- {y}_{1})}^{2}}\end{align*}$  .

  distributive property

  The product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example,  $\begin{align*}a(b+c)=ab+ac\end{align*}$ .  

  diverge

  If a series grows without bound, it is said to diverge.

  dividend

  In a division problem, the dividend is the number or expression that is being divided.

  divisor

  In a division problem, the divisor is the number or expression that is being divided into the dividend. For example, in the expression  $\begin{align*}152\div6\end{align*}$ , 6 is the divisor and 152 is the dividend.

  domain

  The set of $\begin{align*}x\end{align*}$ -values, or input values, for which the function is defined.

  domain restriction

  Limiting the domain of a function to a part of the graph that is one-to-one in order to find an inverse function.

  double right-circular cone

  Two right circular cones joined at the vertex of each cone.

  dummy variable

  A variable that is a placeholder for another expression. We often use "u" as a dummy variable.

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  element

  An object in a set.

  elimination

  The elimination method for solving a system of two equations involves combining the two equations to produce one equation in one variable.

  ellipse

  A conic section that looks like an elongated circle. An ellipse represents all locations in two dimensions that are the same distance from two specified points called foci.  

  empty set

  A set with no elements.

  end behavior

  A description of the trend of a function as input values become very large or very small, represented as the "ends" of a graphed function.

  endpoint

  A point at one end of a line segment or ray.

  equal

  Two expressions that are the same.

  equal sets

  Two sets that contain the same elements.

  equation

  A mathematical sentence that describes two equal quantities. Equations contain equals signs.

  equidistant

  Two or more distances that are equal.

  equivalent fractions

  Fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.

  equivalent sets

  Two sets that have the same cardinality.

  Euler number

  The irrational number $\begin{align*}e\end{align*}$ , such that as $\begin{align*}n \rightarrow \infty, \left(1+ \frac{1}{n}\right)^n \rightarrow e\end{align*}$ . $\begin{align*}e\approx 2.71828\end{align*}$ .

  exponent

  Exponents are indicated by superscript numbers. If the exponent is a positive integer, it is used to describe the number of times that a term is multiplied by itself.

  exponential decay

  Exponential decay occurs when a quantity decreases by the same proportion in each given time period.

  exponential form

  The exponential form of an expression is $\begin{align*}{b}^{x} = a\end{align*}$ , where  $\begin{align*}b\end{align*}$  is the base and  $\begin{align*}x\end{align*}$  is the exponent.

  exponential function

  A function whose variable is in the exponent. The general form is $\begin{align*}y=a\cdot {b}^{x-h} + k\end{align*}$ .

  exponential growth

  Exponential growth occurs when a quantity increases by the same proportion in each given time period.

  expression (algebraic or variable)

  An expression that includes variables. 

  expression (mathematical)

  A mathematical phrase containing variables, operations, and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.

  extraneous solution

  A solution of an equation that, when checked in the equation, is not actually a solution.

  extrema

  A collective term encompassing both the minimum and maximum values, referring to the "extreme" values of the function.