chapters 2 - 7

Function

Function

is a relation that assigns to every element in one set, called the domain exactly one element in another set, called the range

Domain & Range

It can be determined by examining each representation

• if any value in a domain corresponds to more than one value in the rain the relation is not a function

Vertical line test

one way to determine whether a relation is a function. If any vertical line intersects the graph at any point, the relation is not a function

Function notion

represents an output value in terms of an input value. In f(x), x is the independent variable and f(x), also known as y, is the dependent variable

Translation

A vertical and/or horizontal displacement. This transformation shifts the graph without changing the orientation or shape of the graph

• scaled, translated function:

  y= a(x-h) ^2 +k

reflection

flips the graph of the parent function over a line

vertical stretch or compression

changes the shape of the graph by multiplying all of the y values by a factor

relation

is a set of pairs of input and output values. You can represent a relation in four different ways: ordered pairs, mapping diagram, table, and graph

Dependent and Independent

To model real-world situations using a function rule, you need to identify the dependent and independent qualities.

EX. “Distance is a function of time” means distance depends on the time

Origin

The graph of a direct variation function is always a line through the origin usually (0,0)

Slope

The slope of a non-vertical line through points (x1, y1) and (x2, y2) is the ratio of the vertical change to the corresponding horizontal change.

• slope = vertical change (rise) (y2 -y1)

  ----------------------- = ----------

  Horizontal change (run (x2 -x1)

slope form intercept

The slope form intercept form of an equation of a line is y = mx +b where m is the slope of the line and (0, b) is the y-intercept

Standard form

another form of the equations of a line standard form, in which the use of x and y terms are set to a constant. When possible, you write the coefficients of x and y and the constant term as integers.

a standard form of a linear equation is Ax+ By = C, where A, B and C are real numbers and A and B are not both Zero

Point-Slope form

the equation of a line in point-slope form through the point (x1, y1) with slope m: y-y1=m(x-x1)

Parent Function

A parent function is the simplest form in a set of functions that form a family. Each family in the actual family is a transformation of the parent function.

translation

a translation shifts the graph of the -parent function horizontally, vertically, or both without changing the shape orientation.

Reflection

A reflection flips the graph of a function across a line, such as the x and the y axis. each point on the graph of the related function is the same distance from the line of reflection as its corresponding point on the graph of the original function.

vertical stretch

A vertical stretch multiple all y values of a function by the same factor greater than 1. For a function f(x) and a constant a, y - af(x) is a vertical when a >1.

vertical compression

reduces all y values of a function by the same factor between 0 and 1. and a vertical compression when 0 <1/a< 1

linear inequality

A linear inequality is an inequality in two variables whose graph is a region of the coordinate plane bounded by a line.

A solid boundary line means the line is a part of the solution ex. </ >/

A dashed boundary line means the line is not part of the solution < >

substitution

you substitution the x value with the y value into another equation

elimination

It uses the addition property of equality to solve the system.

inconsistent system

an inconsistent system has no solutions

consistent system

A consistent system has at least one solution. It can be:

-an independent system that has one solution

-a dependent system which has infinitely many solutions