### Key Terms

- arrow notation
- a way to represent symbolically 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, that opens up or down, 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

- 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.

- 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

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

- 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 $\left(h,\phantom{\rule{0.8em}{0ex}}\text{}k\right)$ 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 $f(x)={a}_{n}{x}^{n}+\mathrm{...}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$

- 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