Glossary

annuity | An annuity is an investment that is a sequence of equal periodic deposits.

arithmetic sequence | An arithmetic sequence is a sequence where the difference between consecutive terms is constant.

asymptote | A line which a graph of a function approaches closely but never touches.

binomial | A binomial is a polynomial with exactly two terms.

boundary line | The line with equation $Ax+By=C$ is the boundary line that separates the region where $Ax+By>C$ from the region where $Ax+By<C$.

break-even point | The point at which the revenue equals the costs is the break-even point; $C(x)=R(x)$.

circle | A circle is all points in a plane that are a fixed distance from a fixed point in the plane.

coincident lines | Coincident lines have the same slope and same $y$-intercept.

common difference | The difference between consecutive terms in an arithmetic sequence, $a_n−a_{n−1}$, is $d$, the common difference, for $n$ greater than or equal to two.

common logarithmic function | The function $f(x)=\log{x}$ is the common logarithmic function with base10, where $x>0$. $$y=\log{x} \text{ is equivalent to } x=10^y$$

common ratio | The ratio between consecutive terms in a geometric sequence, $\frac{a_n}{a_{n−1}}$, is $r$, the common ratio, where $n$ is greater than or equal to two.

complementary angles | Two angles are complementary if the sum of the measures of their angles is $90$ degrees.

complex conjugate pair | A complex conjugate pair is of the form $a+bi, a-bi$

complex number | A complex number is of the form $a+bi$, where $a$ and $b$ are real numbers. We call $a$ the real part and $b$ the imaginary part.

complex number system | The complex number system is made up of both the real numbers and the imaginary numbers.

complex rational expression | A complex rational expression is a rational expression in which the numerator and/or denominator contains a rational expression.

compound inequality | A compound inequality is made up of two inequalities connected by the word “and” or the word “or.”

conditional equation | An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.

conjugate pair | A conjugate pair is two binomials of the form $(a−b), (a+b)$. The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.

consistent and inconsistent systems | Consistent system of equations is a system of equations with at least one solution; inconsistent system of equations is a system of equations with no solution.

contradiction | An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.

cost function | The cost function is the cost to manufacture each unit times $x$, the number of units manufactured, plus the fixed costs; $C(x) = (\text{cost per unit})x+ \text{fixed costs}$.

critical point of a rational inequality | The critical point of a rational inequality is a number which makes the rational expression zero or undefined.

degree of a constant | The degree of any constant is $0$.

degree of a polynomial | The degree of a polynomial is the highest degree of all its terms.

degree of a term | The degree of a term is the sum of the exponents of its variables.

degree of the polynomial equation | The degree of the polynomial equation is the degree of the polynomial.

determinant | Each square matrix has a real number associated with it called its determinant.

discriminant | In the Quadratic Formula, $x=\frac{-b±\sqrt{b^2-4ac}}{2a}$, the quantity $b^2-4ac$ is called the discriminant.

domain of a relation | The domain of a relation is all the $x$-values in the ordered pairs of the relation.

ellipse | An ellipse is all points in a plane where the sum of the distances from two fixed points is constant.

exponential function | An exponential function, where $a>0$ and $a≠1$, is a function of the form $f(x)=a^x$.

extraneous solution to a rational equation | An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.

factoring | Splitting a product into factors is called factoring.

finite sequence | A sequence with a domain that is limited to a finite number of counting numbers.

function | A function is a relation that assigns to each element in its domain exactly one element in the range.

general term of a sequence | The general term of the sequence is the formula for writing the $n$th term of the sequence. The $n$th term of the sequence, $a_n$, is the term in the $n$th position where $n$ is a value in the domain.

geometric sequence | A geometric sequence is a sequence where the ratio between consecutive terms is always the same

greatest common factor | The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

horizontal line | A horizontal line is the graph of an equation of the form $y=b$. The line passes through the y-axis at $(0,b)$.

hyperbola | A hyperbola is defined as all points in a plane where the difference of their distances from two fixed points is constant.

identity | An equation that is true for any value of the variable is called an Identity. The solution of an identity is all real numbers.

imaginary unit | The imaginary unit $i$ is the number whose square is $–1$. $i^2 = -1$ or $i=\sqrt{-1}$.

infinite geometric series | An infinite geometric series is an infinite sum infinite geometric sequence.

infinite sequence | A sequence whose domain is all counting numbers and there is an infinite number of counting numbers.

intercepts of a line | The points where a line crosses the $x$-axis and the $y$-axis are called the intercepts of the line.

like radicals | Like radicals are radical expressions with the same index and the same radicand.

linear equation | A linear equation is an equation in one variable that can be written, where $a$ and $b$ are real numbers and $a≠0$, as $ax+b=0$.

linear equation | An equation of the form $Ax+By=C$, where $A$ and $B$ are not both zero, is called a linear equation in two variables.

linear inequality | A linear inequality is an inequality that can be written in one of the following forms: $Ax+By>C$, $Ax+By≥C$, $Ax+By<C$, or $Ax+By≤C$, where $A$ and $B$ are not both zero.

logarithmic function | The function $f(x)=\log_a{x}$ is the logarithmic function with base $a$, where $a>0$, $x>0$, and $a≠1$. $$y=\log_a{x} \text{ is equivalent to } x=a^y$$

mapping | A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range.

matrix | A matrix is a rectangular array of numbers arranged in rows and columns.

minor of an entry in a $3×3$ determinant | The minor of an entry in a $3×3$ determinant is the $2×2$ determinant found by eliminating the row and column in the $3×3$ determinant that contains the entry.

monomial | A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form $ax^m$, where $a$ is a constant and $m$ is a whole number.

natural base | The number $e$ is defined as the value of $(1+\frac{1}{n})^n$, as $n$ gets larger and larger. We say, as $n$ increases without bound, $e≈2.718281827...$

natural exponential function | The natural exponential function is an exponential function whose base is $e$: $f(x)=e^x$. The domain is $(−∞,∞)$ and the range is $(0,∞)$.

natural logarithmic function | The function $f(x)=\ln(x)$ is the natural logarithmic function with base $e$, where $x>0$. $$y=\ln{x} \text{ is equivalent to } x=e^y$$

one-to-one function | A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each $y$-value is matched with only one $x$-value.

ordered pair | An ordered pair, $(x,y)$ gives the coordinates of a point in a rectangular coordinate system. The first number is the $x$-coordinate. The second number is the $y$-coordinate.

origin | The point $(0,0)$ is called the origin. It is the point where the $x$-axis and $y$-axis intersect.

parabola | A parabola is all points in a plane that are the same distance from a fixed point and a fixed line.

parallel lines | Parallel lines are lines in the same plane that do not intersect.

partial sum | When we add a finite number of terms of a sequence, we call the sum a partial sum.

perpendicular lines | Perpendicular lines are lines in the same plane that form a right angle.

point-slope form | The point-slope form of an equation of a line with slope $m$ and containing the point $(x_1,y_1)$ is $y−y_1=m(x−x_1)$.

polynomial | A monomial or two or more monomials combined by addition or subtraction is a polynomial.

polynomial equation | A polynomial equation is an equation that contains a polynomial expression.

polynomial function | A polynomial function is a function whose range values are defined by a polynomial.

Power Property | According to the Power Property, $a$ to the $m$ to the $n$ equals $a$ to the $m$ times $n$.

Product Property | According to the Product Property, $a$ to the $m$ times $a$ to the $n$ equals $a$ to the $m$ plus $n$.

Product to a Power | According to the Product to a Power Property, $a$ times $b$ in parentheses to the $m$ equals $a$ to the $m$ times $b$ to the $m$.

Properties of Negative Exponents | According to the Properties of Negative Exponents, $a$ to the negative $n$ equals $1$ divided by $a$ to the $n$ and $1$ divided by $a$ to the negative $n$ equals $a$ to the $n$.

proportion | When two rational expressions are equal, the equation relating them is called a proportion.

quadratic equation | Polynomial equations of degree two are called quadratic equations.

quadratic function | A quadratic function, where $a$, $b$, and $c$ are real numbers and $a≠0$, is a function of the form $f(x)=ax^2+bx+c$.

quadratic inequality | A quadratic inequality is an inequality that contains a quadratic expression.

Quotient Property | According to the Quotient Property, $a$ to the $m$ divided by $a$ to the $n$ equals $a$ to the $m$ minus $n$ as long as $a$ is not zero.

Quotient to a Negative Exponent | Raising a quotient to a negative exponent occurs when $a$ divided by $b$ in parentheses to the power of negative $n$ equals $b$ divided by $a$ in parentheses to the power of $n$.

Quotient to a Power Property | According to the Quotient to a Power Property, $a$ divided by $b$ in parentheses to the power of $m$ is equal to $a$ to the $m$ divided by $b$ to the $m$ as long as $b$ is not zero.

radical equation | An equation in which a variable is in the radicand of a radical expression is called a radical equation.

radical function | A radical function is a function that is defined by a radical expression.

range of a relation | The range of a relation is all the $y$-values in the ordered pairs of the relation.

rational equation | A rational equation is an equation that contains a rational expression.

rational expression | A rational expression is an expression of the form $\frac{p}{q}$, where $p$ and $q$ are polynomials and $q≠0$.

rational function | A rational function is a function of the form $R(x)=\frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomial functions and $q(x)$ is not zero.

rational inequality | A rational inequality is an inequality that contains a rational expression.

rationalizing the denominator | Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.

relation | A relation is any set of ordered pairs, $(x,y)$. All the $x$-values in the ordered pairs together make up the domain. All the $y$-values in the ordered pairs together make up the range.

revenue | The revenue is the selling price of each unit times $x$, the number of units sold; $R(x) = (\text{selling price per unit})x$.

row-echelon form | A matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a $1$ and all entries below the diagonal are zeros.

sequence | A sequence is a function whose domain is the counting numbers.

similar figures | Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides have the same ratio.

simplified rational expression | A simplified rational expression has no common factors, other than $1$, in its numerator and denominator.

solution of a linear equation in two variables | An ordered pair $(x,y)$ is a solution of the linear equation $Ax+By=C$, if the equation is a true statement when the $x$- and $y$-values of the ordered pair are substituted into the equation.

solution of an equation | A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

solution to a linear inequality | An ordered pair $(x,y)$ is a solution to a linear inequality if the inequality is true when we substitute the values of $x$ and $y$.

solutions of a system of equations | Solutions of a system of equations are the values of the variables that make all the equations true; solution is represented by an ordered pair $(x,y)$.

solutions of a system of linear equations with three variables | The solutions of a system of equations are the values of the variables that make all the equations true; a solution is represented by an ordered triple $(x,y,z)$.

square matrix | A square matrix is a matrix with the same number of rows and columns.

square of a number | If $n^2=m$, then $m$ is the square of $n$.

square root of a number | If $n^2=m$, then $n$ is a square root of $m$.

standard form | A complex number is in standard form when written as $a+bi$, where $a$, $b$ are real numbers.

standard form of a linear equation | A linear equation is in standard form when it is written $Ax+By=C$.

standard form of a polynomial | A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.

supplementary angles | Two angles are supplementary if the sum of the measures of their angles is $180$ degrees.

system of linear equations | When two or more linear equations are grouped together, they form a system of linear equations.

system of linear inequalities | Two or more linear inequalities grouped together form a system of linear inequalities.

system of nonlinear equations | A system of nonlinear equations is a system where at least one of the equations is not linear.

trinomial | A trinomial is a polynomial with exactly three terms.

vertical line | A vertical line is the graph of an equation of the form $x=a$. The line passes through the $x$-axis at $(𝑎,0)$.

Zero Exponent Property | According to the Zero Exponent Property, $a$ to the zero is $1$ as long as $a$ is not zero.

zero of the function | A value of $x$ where the function is $0$, is called a zero of the function.

Zero Product Property | The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero.