3.1 - Complex numbers

Arrow Notation

A method of notation for that is used to show that x or f(x) is approaching a particular value

Term

A single number or variable

Polynomial function

Consists of either zero or the sum of a number of a nonzero terms

Reciprocal Function

Function defined on the set of nonzero reals, that sends every real number to its reciprocal

Remainder theorem

If a polynomial(x) is divided by (x-k) then the remainder is the value of f(k)

Maximum Value

If the parabola opens down the vertex represents the highest point on the graph

Minimum Value

If the parabola opens up, the vertex represents the lowest point on the graph

Rational Zero Theorem

If the polynomial has integer coefficients, then every rational zero of f(x) has the form p/q where p is a factor of the constant term asub(o) and q is a factor of the leading coefficient asub(n)

Fundamental theorem

If you have a polynomial of a degree greater than 0, then f(x) has at least one complex zero. We can use when arguing if f(x) is a polynomial degree n>0 and "a" is a non-zero real number then f(x) has exactly n linear factors

Rational Function

Is a function that can be written as the quotient of two polynomial functions

Power Function

Is a function with a single term that is the product of a real number

Turning point

Is a point on a graph that changes the direction from increasing to decreasing or decreasing to increasing

Coefficent

Is a variable raised to a fixed number

Complex Number

Is the sum of a real number and an imaginary number. is expressed in standard form when written in a+bi

Leading term

Is the term containing the highest power of the variable- term with the highest degree

Factor theorem

K is a zero of f(x) and only if (x-k) is a factor of f(x)

Horizontal Asymptote

Line along y-axis that helps describe the behavior of a graph as the input gets very large or very small. Horizontal line that the graph approaches as the input increases or decreases without bound

Continuous function

Meaning has no breaks in a graph - continuous flow

Vertex

One important feature of the graph is that it has an extreme point

Vertical asymptote

Straight lines along the x-axis that can be determined using one sided limits/describe the behavior of a graph as the output gets large/ line that the graph approaches but never crosses

Axis of Symmetry

The graph is also symmetric with a vertical line drawn through the vertex

Parabola

The graph of a quadratic function is a v-shaped curve called a parabola

Degree

The highest power of the variable that occurs in the polynomial

Descartes Rule of Signs

The number of positive real zeros is either equal to the number of sign changes of f(x) or is less than the number of sign changes by an even #

Removable discontinuity

When a graph contains a hole; a single point where the graph is not defined, indicated by an open circle

Complex Conjugate

When dividing a- it can be found by changing the sign of the imaginary part of the complex number ( real part is left unchanged)

Complex Plane

a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary components

Imaginary Number

defined as the square root of (-1)

Multiplicity

the number of times a given factor appears in the factored form of the equation of a polynomial

Direct variation

variation where y=kx^n where k is the constant of proportionality or constant of variatio; Variation where Both variables react the same way

Zeros/roots

x-intercept are the points at which the parabola crosses the x-axis . If they exist, the x-intercepts represent the zeros or roots of the quadratic function ( values of x)

Joint Variation

y=kx/z or y=kxz where k is the constant of proportionality or constant of variation; where one variable is directly or inversely related to more than one variable