Key Terms

arrow notation

a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value

axis of symmetry

a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by $x=-\frac{b}{2a}.$

coefficient

a nonzero real number multiplied by a variable raised to an exponent

complex conjugate

the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number

complex number

the sum of a real number and an imaginary number, written in the standard form $$a+bi,$$ where $a$ is the real part, and $bi$ is the imaginary part

complex plane

a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number

constant of variation

the non-zero value $k$ that helps define the relationship between variables in direct or inverse variation

continuous function

a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph

degree

the highest power of the variable that occurs in a polynomial

Descartes’ Rule of Signs

a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of $f(x)$ and $f(-x)$

direct variation

the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other

Division Algorithm

given a polynomial dividend $f(x)$ and a non-zero polynomial divisor $d(x)$ where the degree of $d(x)$ is less than or equal to the degree of $f(x),$ there exist unique polynomials $q(x)$ and $r(x)$ such that $f(x)=d(x)q(x)+r(x)$ where $q(x)$ is the quotient and $r(x)$ is the remainder. The remainder is either equal to zero or has degree strictly less than $d(x).$

end behavior

the behavior of the graph of a function as the input decreases without bound and increases without bound

Factor Theorem

$k$ is a zero of polynomial function $f(x)$ if and only if $(x-k)$ is a factor of $f(x)$

Fundamental Theorem of Algebra

a polynomial function with degree greater than 0 has at least one complex zero

general form of a quadratic function

the function that describes a parabola, written in the form $f(x)=a{x}^2+bx+c,$ where $a,b,$ and $c$ are real numbers and $a\ne 0.$

global maximum

highest turning point on a graph; $f(a)$ where $f(a)\ge f(x)$ for all $x.$

global minimum

lowest turning point on a graph; $f(a)$ where $f(a)\le f(x)$ for all $x.$

horizontal asymptote

a horizontal line $y=b$ where the graph approaches the line as the inputs increase or decrease without bound.

imaginary number

a number in the form $bi$ where $i=\sqrt{-1}$

Intermediate Value Theorem

for two numbers $a$ and $b$ in the domain of $f,$ if $a<b$ and $f\left(a\right)\ne f\left(b\right),$ then the function $f$ takes on every value between $f\left(a\right)$ and $f\left(b\right);$ specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the $x\text{-}$ axis

inverse variation

the relationship between two variables in which the product of the variables is a constant

inversely proportional

a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases

invertible function

any function that has an inverse function

joint variation

a relationship where a variable varies directly or inversely with multiple variables

leading coefficient

the coefficient of the leading term

leading term

the term containing the highest power of the variable

Linear Factorization Theorem

allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form $(x-c),$ where $c$ is a complex number

multiplicity

the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form ${(x-h)}^p,$ $x=h$ is a zero of multiplicity $p.$

polynomial function

a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

power function

a function that can be represented in the form $f(x)=k{x}^p$ where $k$ is a constant, the base is a variable, and the exponent, $p,$ is a constant

rational function

a function that can be written as the ratio of two polynomials

Rational Zero Theorem

the possible rational zeros of a polynomial function have the form $\frac{p}{q}$ where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Remainder Theorem

if a polynomial $f(x)$ is divided by $x-k,$ then the remainder is equal to the value $f(k)$

removable discontinuity

a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function

smooth curve

a graph with no sharp corners

standard form of a quadratic function

the function that describes a parabola, written in the form $f(x)=a{(x-h)}^2+k,$ where $\left(h,k\right)$ is the vertex.

synthetic division

a shortcut method that can be used to divide a polynomial by a binomial of the form $x-k$

term of a polynomial function

any $a_i{x}^i$ of a polynomial function in the form $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_2{x}^2+a_{1}x+a_0$

turning point

the location at which the graph of a function changes direction

varies directly

a relationship where one quantity is a constant multiplied by the other quantity

varies inversely

a relationship where one quantity is a constant divided by the other quantity

vertex

the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function

vertex form of a quadratic function

another name for the standard form of a quadratic function

vertical asymptote

a vertical line $x=a$ where the graph tends toward positive or negative infinity as the inputs approach $a$

zeros

in a given function, the values of $x$ at which $y=0,$ also called roots